Extended implicative groupoids

Abstract

In this paper, we generalize the concept of extended order algebras in order to get a new algebraic structure called “extended implicative groupoid”. First, we define the notions of pre-weak extended, weak extended, right extended and left extended implicative groupoid. Then we introduce the concept of extended implicative groupoid by using these notions. In addition, the special properties of these structures, such as the existence of MacNeille completion and adjoint product are studied. Finally, we prove that the class of symmetrical associative complete distributive extended implicative groupoids, coincides with complete residuated lattices.

Keywords

Extended order algebra extended implicative groupoid MacNeille completion residuated lattice 06F99 18B40

1 Introduction

Residuated structures, as an important family of ordered algebras, have been studied by many mathematicians and have many applications in mathematics and logic [1–7 , 15]. For the sake of presenting an order theoretical investigation of the algebras in logics which is concerned with implication structures, Rasiowa [16] introduced implicative algebras and Guido and Toto [8, 13] recently investigated and renamed them as extended order algebras. An implicative algebra (or extended order algebra) is a generalization of some ordered set with the greatest element which were considered in some studies [12, 13]. Guido (TACL-2013) suggested that some more general structures are obtained by weakening the definition of an extended order algebra and replacing the greatest element by some arbitrary subset, which were named “weak implicative groupoids” by Guido [11]. He claimed that these structures have great applications which are more crucial for logic related purposes.

In this paper, the idea of Guido is followed, after introducing the class of weak extended implicative groupoids. Then the special properties of these structures are considered. When (L, → , E) is a symmetrical associative complete distributive extended implicative groupoid with the adjoint product ⊗ and dual (L, ⇝ , E) such that (E, ⪯) attains its greatest lower bound ⊥_E ∈ E, then (L, ≤ , ⊗ , ⊥ _E) satisfies all of the properties in complete residuated lattices. Hence we see that extended implicative groupoids are useful algebraic structures which strictly contain extended order algebras, while they satisfy almost all good properties of extended order algebras (including the existence of MacNeille completion), under additional assumptions that the underlying upper bound set E contains both its lower and upper bounds. Hence there would be no (much) loss of generality in replacing the existed singleton {⊤} in the settings of extended order algebras with some non-empty upper bound set E, especially when E attains both its lower and upper bounds. Extended implicative groupoids are natural extensions of extended order algebras and implicative groupoids. Hence the motivation and applications of Extended implicative groupoids more or less, are the same as motivation and applications of extended order algebras and implicative groupoids, which is comprehensively discussed in [8 , 13]. In other words, the concept of extended implicative groupoids is established by notion of order and its extension to many valued case, which can be considered as a kind of suitable many-valued logic systems based on the fundamental connective of implication. As mentioned in [8 , 13], the common base of weak implicative groupoids allows to consider it as a lattice-order structure that is a fundamental requirement for most applications, such as the MacNeille completion. In our work, an important approach is to provide a suitable structure for many valued logic and to introduce its algebraic properties.

2 Preliminaries

In this section, we collect some notions on order structures and related topics which are used in the next sections.

A reflexive and transitive relation ≤ on a set L is called a pre-order. A pre-ordered set (L, ≤) that satisfies an antisymmetry criterion is called an ordered set. Let (L, ≤) be a pre-ordered set and S ⊆ K ⊆ L. We denote the set of all upper bounds and lower bounds of S in the pre-ordered set (K, ≤) by $U_{K} (S)$ and $L_{K} (S)$ , respectively, as follows $\begin{matrix} U_{K} (S) = {x \in K : a \leq x, \forall a \in S}, \\ L_{K} (S) = {x \in K : x \leq a, \forall a \in S} . \end{matrix}$ When K = L, we simply use $U (S)$ and $L (S)$ . If S = {a}, we use ↑a and ↓a, instead of $U ({a})$ and $L ({a})$ , respectively. In addition, we use the notions ∨S and ∧S to represent the set of all least upper bounds and greatest lower bounds of S in L, respectively. For a binary operation → : L × L ⟶ L and A, B ⊆ L, we set A → B = {a → b : a ∈ A, b ∈ B}.

Definition 2.1. [13, Definition 5] A weak extended order algebra is a triple (L, → , ⊤), where L is a non-empty set, → is a binary operation on L, ⊤ is a distinguished element of L such that the following assertions are satisfied for all a, b, c ∈ L: (o₁) a→ ⊤ = ⊤,

(o₂) a→ a = ⊤,

(o₃) if a→ b = ⊤ and b→ a = ⊤, then a = b,

(o₄) if a→ b = ⊤ and b→ c = ⊤, then a→ c = ⊤.

A unique natural order with the greatest element ⊤ is related to any weak extended order algebra (L, → , ⊤), that for any a, b ∈ L is defined by a ≤ b if and only if a→ b = ⊤. Conversely, regarding any partially ordered set (L, ≤) with the greatest element ⊤, corresponds a (not necessarily unique) weak extended order algebra (L, → , ⊤), where → : L × L ⟶ L is a ⊤-extension of ≤.

Definition 2.2. [13, Definition 15] A triple (L, → , ⊤) is called an extended order algebra, if it satisfies the axioms (o₁), (o₂) and (o₃) and for all a, b, c ∈ L,

(o₅) if a→ b = ⊤, then (c→ a) → (c → b) = ⊤,

$(o_{5}^{'})$ if a→ b = ⊤, then (b→ c) → (a → c) = ⊤.

In [11], as a generalization of extended order algebras, the concept of a weak implicative groupoid is introduced as follows.

Definition 2.3. [11] A weak implicative groupoid is a triple (L, → , E), where L is a non-empty set, → indicates a binary operation on L and ∅ ≠ E ⊊ L, so that the following conditions are satisfied for all a, b, c ∈ L:

(e₁) a → a ∈ E,

(e₂) if a → b ∈ E and b → a ∈ E, then a = b,

(e₃) if a → b ∈ E and b → c ∈ E, then a → c ∈ E.

3 Extended implicative groupoids and some properties

As it seems by Definition 2.3, we should get rid of upper bound condition on whole L, i.e. the condition (o₁), in order to introduce a non-trivial weak implicative groupoid (L, → , E), in such a way that the binary relation induced by → can remain a partial order. Hence weak implicative groupoids do not seem to be a natural extension of weak extended order algebras. In order to reach such a natural extension, we should restrict ourselves to the case where E is a partially ordered set and an upper bound set for L ∖ E. Now, the following definition, as a generalization of weak extended order algebras, is presented as a special subset of weak implicative groupoids.

Notations. Let L be a non-empty set, E be a proper subset of L and → : L × L ⟶ L be a binary operation on L. We named the following axioms:

(wei1) λ → e ∈ E, for any λ ∈ L ∖ E and e ∈ E,

(wei2) a → a ∈ E, for any a ∈ L,

(wei3) if a → b ∈ E and b → a ∈ E, then a = b, for any a, b ∈ L,

(wei4) if a → b ∈ E and b → c ∈ E, then a → c ∈ E, for any a, b, c ∈ L,

(pwei) if a → b ∈ E and b → c ∈ E, then a → c ∈ E, for any a, b, c ∈ E,

(rei) if a → b ∈ E, then (c → a) → (c → b) ∈ E, for any a, b, c ∈ L,

(lei) if a → b ∈ E, then (b → c) → (a → c) ∈ E, for any a, b, c ∈ L.

Definition 3.1 Let L be a non-empty set, E be a proper subset of L and → be a binary operation on L. Then (L, → , E) is called:

(i) A pre-weak extended implicative groupoid, if it satisfies (wei1)-(wei3) and (pwei),

(ii) A weak extended implicative groupoid, if it satisfies (wei1)-(wei4),

(iii) A right extended implicative groupoid, if it satisfies (wei1)-(wei3) and (rei),

(iv) A left extended implicative groupoid, if it satisfies (wei1)-(wei3) and (lei),

(v) An extended implicative groupoid, if it is a right and a left extended implicative groupoid.

We show that (wei4) is absolutely stronger that (pwei) and the axioms (wei1), (wei2), (wei3), (rei), (lei) and (either (wei4) or (pwei)) are mutually independent.

Example 3.2 (i) Obviously (wei4) implies (pwei). Let L = {a, b, c, d, e} and E = {c, d, e}. We define the operation → on L as follows:

→	a	b	c	d	e
a	a	c	a	a	a
b	a	a	c	a	a
c	a	a	c	c	c
d	a	a	a	c	c
e	a	a	a	a	c

Then (L, → , E) satisfies (pwei) but does not satisfy (wei4), since b → c, c → e ∈ E but b → e ∉ E.

(ii) Let L₁ = {a, b, c, d} and E₁ = {c, d}. We define the operation →₁ on L as follows:

→₁	a	b	c	d
a	c	a	a	a
b	a	c	b	b
c	a	a	c	a
d	a	a	a	c

Then (L₁, → ₁, E₁) satisfies (wei2), (wei3), (wei4), (pwei), (rei) and (lei) but does not satisfy (wei1), since a → ₁c ∉ E₁.

(iii) Let L₂ = {a, b, c, d, e} and E₂ = {c, d, e}. We define the operation →₂ on L as follows:

→₂	a	b	c	d	e
a	c	a	c	c	c
b	a	a	c	c	c
c	a	a	c	a	a
d	a	a	a	a	a
e	a	a	a	a	a

Then (L₂, → ₂, E₂) satisfies (wei1), (wei3), (wei4), (pwei), (rei) and (lei) but does not satisfy (wei2), since b → ₂b ∉ E₂.

(iv) Let L₃ = {a, b, c, d, e} and E₃ = {c, d, e}. We define the operation →₃ on L as follows:

→₃	a	b	c	d	e
a	c	c	c	c	c
b	c	c	c	c	c
c	a	a	c	a	a
d	a	a	a	c	a
e	a	a	a	a	c

Then (L₃, → ₃, E₃) satisfies (wei1), (wei2), (wei4), (pwei), (rei) and (lei) but does not satisfy (wei3), since a → ₃b, b → ₃a ∈ E₃ but a ≠ b.

(v) Let L₄ = {a, b, c, d, e} and E₄ = {d, e}. We define the operation →₄ on L as follows:

→₄	a	b	c	d	e
a	d	d	a	d	d
b	a	d	d	d	d
c	a	a	d	d	d
d	a	a	a	d	a
e	a	a	a	a	d

Then (L₄, → ₄, E₄) satisfies (wei1), (wei2), (wei3), (pwei), (rei) and (lei) but does not satisfy (wei4), since a → ₄b, b → ₄c ∈ E₄ but a → ₄c ∉ E₄.

(vi) Let L₅ = {a, b, c, d, e} and E₅ = {c, d, e}. We define the operation →₅ on L as follows:

→₅	a	b	c	d	e
a	c	a	c	c	c
b	a	c	c	c	c
c	a	a	c	c	a
d	a	a	a	c	c
e	a	a	a	a	c

Then (L₅, → ₅, E₅) satisfies (wei1), (wei2), (wei3), (rei) and (lei) but does not satisfy (pwei), since c → ₅d, d → ₅e ∈ E₅ but a → ₅c ∉ E₅.

(vii) Let L₆ = {a, b, c, d, e} and E₆ = {d, e}. We define the operation →₆ on L as follows:

→₆	a	b	c	d	e
a	d	d	a	d	d
b	a	d	a	d	d
c	b	a	d	d	d
d	a	a	a	d	a
e	a	a	a	a	d

Then (L₆, → ₆, E₆) satisfies (wei1), (wei2), (wei3), (wei4), (pwei) and (lei) but does not satisfy (rei), since a → ₆b ∈ E₆ but (c → ₆a) → ₆ (c → ₆b) = a ∉ E₆.

(viii) Let L₇ = {a, b, c, d, e} and E₇ = {d, e}. We define the operation →₇ on L as follows:

→₇	a	b	c	d	e
a	d	a	a	d	d
b	d	d	b	d	d
c	a	a	d	d	d
d	a	a	a	d	a
e	a	a	a	a	d

Then (L₇, → ₇, E₇) satisfies (wei1), (wei2), (wei3), (wei4), (pwei) and (rei) but does not satisfy (lei), since b → ₇d ∈ E₇ but (d → ₇c) → ₇ (b → ₇c) = a ∉ E₇.

Now, we give some relations among these structures.

Proposition 3.3 (i) Any weak extended order algebra is a weak extended implicative groupoid.

(ii) Any weak extended implicative groupoid is a weak implicative groupoid.

(iii) Any right extended implicative groupoid is a weak extended implicative groupoid.

(iv) Any left extended implicative groupoid is a weak extended implicative groupoid.

(v) Any extended implicative groupoid is a weak extended implicative groupoid.

(vi) Any extended order algebra is an extended implicative groupoid, and any extended implicative groupoid is a weak implicative groupoid.

Proof. (i) Let (L, → , ⊤) be a weak extended order algebra. Then all of the axioms (wei1)-(wei4) are satisfied by setting E = {⊤}. Hence (L, → , ⊤) is a weak extended implicative groupoid.

(ii) Let (L, → , E) be a weak extended implicative groupoid. Then (L, → , E) is a weak implicative groupoid based on (wei2)-(wei4). (iii) Let (L, → , E) be a right extended implicative groupoid. It is enough to show that (rei) implies (wei4). For this suppose, we have a → b ∈ E and b → c ∈ E. By (rei), we have (a → b) → (a → c) ∈ E. Since a → b ∈ E, we have a → c ∈ E by (wei1) and (wei3). Hence (wei4) holds.

(iv) The proof is similar to the proof of (iii).

(v) , (vi) The proofs are clear.□

Using proposition 3.3, the relationship between extended order algebras (Eoa), right extended implicative groupoids (Reig), extended implicative groupoids (Eig), weak extended order algebras (Weoa), weak extended implicative groupoids (Weig), weak implicative groupoids (Wig) and left extended implicative groupoids (Leig) can be described by the following diagram:

We show that the converse of statements (i) - (vi) in Proposition 3.3, does not hold in general.

Example 3.4 (i) Let L₁ = {a, b, c, d}, E₁ = {c, d} and the operation →₁ on L is defined as follows:

→₁	a	b	c	d
a	c	c	c	c
b	a	d	c	c
c	a	a	d	c
d	a	a	a	c

Then (L₁, → ₁, E₁) is a weak extended implicative groupoid but it is not a weak extended order algebra, since we can not find a unique element in L₁ that plays the role of ⊤ in the settings of a weak extended order algebra.

(ii) Let L₂ = {a, b, c, d}, E₂ = {c, d} and the operation →₂ on L is defined as follows:

→₂	a	b	c	d
a	c	c	a	a
b	a	d	a	a
c	a	a	d	c
d	a	a	a	c

Then (L₂, → ₂, E₂) is a weak implicative groupoid but it is not a weak extended implicative groupoid, since a → ₂c, a → ₂d ∉ E₂ (that is, (wei1) does not hold).

(iii) Let L₃ = {a, b, c, d}, E₃ = {c, d} and the operation →₃ on L is defined as follows:

→₃	a	b	c	d
a	c	a	c	c
b	a	c	c	c
c	a	a	c	a
d	a	a	b	c

Then (L₃, → ₃, E₃) is a weak extended implicative groupoid but it is not a right implicative groupoid, since a → ₃c ∈ E₃ but $(d \to_{3} a) \to_{3} (d \to_{3} c) = a \to_{3} b = a \notin E_{3} .$ (that is, (rei) does not hold).

(iv) Let L₄ = {a, b, c, d}, E₄ = {c, d} and the operation →₄ on L is defined as follows:

→₄	a	b	c	d
a	c	a	c	c
b	a	c	c	c
c	a	a	c	d
d	a	a	a	c

Then (L₄, → ₄, E₄) is a weak extended implicative groupoid but it is not a left implicative groupoid, since a → ₄c ∈ E₄ but $(c \to_{4} d) \to_{4} (a \to_{4} d) = d \to_{4} c = a \notin E_{4} .$ (that is, (lei) does not hold).

(v) Both of (L₃, → ₃, E₃) and (L₄, → ₄, E₄) are weak extended implicative groupoids but neither of them is an extended implicative groupoid, since (rei) and (lei) do not hold, respectively.

(vi) Let L₅ = {a, b, c, d}, E₅ = {c, d} and the operation →₅ on L is defined as follows:

→₅	a	b	c	d
a	c	a	c	d
b	a	c	c	d
c	a	a	c	d
d	a	a	a	d

Then (L₅, → ₅, E₅) is an extended implicative groupoid but it is not an extended order algebra, since we can not find a unique element in L₅ that plays the role of ⊤ in the settings of an extended order algebra.

(vii) Let L₆ = {a, b, c, d}, E₆ = {c, d} and the operation →₆ on L is defined as follows:

→₆	a	b	c	d
a	c	a	a	a
b	a	c	a	a
c	a	a	c	a
d	a	a	a	c

Then (L₆, → ₆, E₆) is a weak implicative groupoid but it is not an extended implicative groupoid, since a → ₆c ∉ E (that is, (wei1) does not hold).

Notation 3.5 (i) Let (L, → , E) be a weak extended implicative groupoid and a, b ∈ L. We define the binary relation ≤ on L as follows: $a \leq b if and only if a \to b \in E .$ Then it is easy to see that (L, ≤) is a poset, and E is an upper bound of L ∖ E respect to ≤.

(ii) Let (L, → , E) be a pre-weak extended implicative groupoid and a, b ∈ E. Define the binary relation ⪯ on E as follows: $a ⪯ b if and only if a \to b \in E .$ Then it is easy to see that (E, ⪯) is a poset.

Definition 3.6 The triple (L, → , E) is called a complete (weak) extended implicative groupoid, if it is a (weak) extended implicative groupoid and (L, ≤) is a complete lattice.

By using the ordered structure of E, the following definition introduces a property which is stronger than transitivity.

Definition 3.7 A pre-weak extended implicative groupoid (L, → , E) is called compatible, if it satisfies in the following condition:

(cpwei) a → b ∈ E if and only if (a → e) → (b → e) ∈ E, for any e ∈ E and a, b ∈ L ∖ E.

Proposition 3.8 Any compatible pre-weak extended implicative groupoid (L, → , E) is a weak extended implicative groupoid.

Proof. Since (L, → , E) is a pre-weak extended implicative groupoid, the conditions (wei1)-(wei3) hold. Let a, b, c ∈ L such that a → b ∈ E and b → c ∈ E. If either a, b or c is in E, the transitivity condition (wei4) holds since (E, ⪯) is a poset. If a ∈ E, then b, c ∈ E, and a ⪯ b ⪯ c, from a → b ∈ E and b → c ∈ E, which implies that a ⪯ c or equivalently a → c ∈ E. If b ∈ E, then c ∈ E and two possibilities may occur. If a ∈ E, then b, c ∈ E, and a ⪯ b ⪯ c, which implies that a → c ∈ E. If a ∈ L ∖ E, since c ∈ E, then by (wei1) a → c ∈ E holds. Finally, if c ∈ E, then by a similar argument, the desired result follows. Hence it is enough to assume a, b, c ∈ L ∖ E. By (cpwei), for all e ∈ E, (a → e) → (b → e) ∈ E and (b → e) → (c → e) ∈ E. Since a → e, b → e, c → e ∈ E and a → e ⪯ b → e ⪯ c → e, we get that (a → e) → (c → e) ∈ E. Hence by (cpwei), we have a → c ∈ E.□

We show that the converse of Proposition 3.8, does not hold in general.

Example 3.9 Let L = {a, b, c, d}, E = {c, d} and the operation → on L is defined as follows:

→	a	b	c	d
a	c	c	c	c
b	a	c	d	c
c	a	a	c	a
d	a	a	a	c

Then (L, → , E) is a (weak) extended implicative groupoid but it is not a compatible pre-weak extended implicative groupoid, since a → b = c ∈ E but $(a \to c) \to (b \to c) = c \to d = a \notin E .$

Definition 3.10 Let (L, → , E) be a weak extended implicative groupoid. The triple (L, → , E) is called:

(i) A left distributive extended implicative groupoid, if it satisfies the left distributivity condition as follows:

(ldei) If $U (A) = U (A^{'})$ , then $L (A \to B) = L (A^{'} \to B)$ , for any A, A′, B ⊆ L.

(ii) A right distributive extended implicative groupoid, if it satisfies the right distributivity condition as follows:

(rdei) If $L (B) = L (B^{'})$ , then $L (A \to B) = L (A \to B^{'})$ , for any A, B, B′ ⊆ L.

(iii) A distributive extended implicative groupoid, if it satisfies the distributivity condition as follows:

(dei) If $U (A) = U (A^{'})$ and $L (B) = L (B^{'})$ , then $L (A \to B) = L (A^{'} \to B^{'})$ , for any A, A′, B, B′ ⊆ L.

Proposition 3.11 (i) Any (left/ right) distributive extended implicative groupoid is a (left/ right) extended implicative groupoid.

(ii) Let (L, → , E) be an extended implicative groupoid and consider the following conditions for any A, B ⊆ L:

(Δ) (⋁ A) → (⋀ B) = ⋀ (A → B),

(lΔ) (⋁ A) → B = ⋀ (A → B),

(rΔ) A → (⋀ B) = ⋀ (A → B).

Then (dei) ⇔ (Δ), (ldei) ⇔ (lΔ) and (rdei) ⇔ (rΔ).

Proof. (i) The proof is similar to that of [13, Proposition 18].

(ii) The proof is similar to [13, Proposition 21]. For instance, we prove (ldei) ⇒ (lΔ). Suppose (ldei) holds. Since $U (A) = U (⋁ A)$ , for arbitrary B ⊆ L, by (ldei), we have $L (A \to B) = L ((⋁ A) \to B)$ . Thus (⋁ A) → B = ⋀ (A → B).□

Definition 3.12 Let (L, → , E) be a weak extended implicative groupoid. The triple (L, → , E) is called:

(i) Lower set preserving, if it satisfies the condition,

(lpei) if $L (A) = L (A^{'})$ and $L (B) = L (B^{'})$ , then $L (A \to B) = L (A^{'} \to B^{'})$ , for any A, A′, B, B′ ⊆ L.

(ii) Upper set preserving, if it satisfies the condition,

(upei) if $U (A) = U (A^{'})$ and $U (B) = U (B^{'})$ , then $U (A \to B) = U (A^{'} \to B^{'})$ , for any A, A′, B, B′ ⊆ L.

Proposition 3.13 Let (L, → , E) be a weak extended implicative groupoid. Then both of (lpei) and (upei) imply (rei).

Proof. Suppose (lpei) holds and a, b ∈ L such that a → b ∈ E. Let c ∈ L such that A = {c} = A′, B = {a} and B′ = {a, b}. Then $L (A) = ↓ c = L (A^{'})$ and $L (B) = ↓ a = L (B^{'})$ . Consequently, $\begin{matrix} L (A \to B) & = L ({c \to a}) = L ({c \to a, c \to b}) \\ = L (A^{'} \to B^{'}) . \end{matrix}$ Particularly, this implies c → a ≤ c → b, which means (rei) holds. Now, suppose (upei) holds and a, b ∈ L such that a → b ∈ E. Let c ∈ L such that A = {c} = A′, B = {b} and B′ = {a, b}. Then $U (A) = ↑ c = U (A^{'})$ and $U (B) = ↑ b = U (B^{'})$ . Consequently, $\begin{matrix} U (A \to B) & = U ({c \to b}) = U ({c \to a, c \to b}) \\ = U (A^{'} \to B^{'}), \end{matrix}$ which implies c → a ≤ c → b. Hence, (rei) holds.

□

Now, we give some examples of the above definitions and their relation.

Example 3.14 (i) Let $ℤ$ be the set of integers, $L = {n ℤ : n = 2 k, k \in ℤ}$ and $E = {2 ℤ, 4 ℤ}$ . Consider the operation → on L is defined as follows: $x \to y = {\begin{matrix} 2 ℤ & x \subseteq y, y = 2 ℤ, \\ 4 ℤ & x \subseteq y, y \neq 2 ℤ, \\ 6 ℤ & x ⊊ y . \end{matrix}$ Then the triple (L, → , E) is a weak extended implicative groupoid.

(ii) Let L = [0, 1] and $E = [\frac{1}{2}, 1]$ be two intervals of real numbers. Consider the operation → on L is defined as follows:

$x \to y = {\begin{matrix} \frac{1}{2} \lor (x + y - 1) & x \leq y, \\ x \land (1 - y) & y < x and x, y \in L ∖ E, \\ y & y < x and x \in E, y \in L ∖ E, \\ 0 & y < x and x, y \in E . \end{matrix}$

Then the triple (L, → , E) is a weak extended implicative groupoid, but it is not a weak extended order algebra, because the upper bound set E of true values is not a singleton. It is worth noting that $\frac{1}{4} \to \frac{1}{3} \in E$ but $(\frac{1}{3} \to \frac{1}{6}) \to (\frac{1}{4} \to \frac{1}{6}) = \frac{1}{3} \to \frac{1}{4} = \frac{1}{3} \notin E .$ Hence (L, → , E) is not a left extended implicative groupoid. Similarly, $\frac{1}{4} \to \frac{3}{4} \in E$ but $(\frac{5}{6} \to \frac{1}{4}) \to (\frac{5}{6} \to \frac{3}{4}) = \frac{1}{4} \to 0 = \frac{1}{4} \notin E .$ Thus (L, → , E) is not a right extended implicative groupoid.

(iii) Let $L = ℕ$ and $E = {2 n : n \in ℕ}$ . We define the operation → on L as follows: $x \to y = {\begin{matrix} 2 & x, y \in E and x \leq y, \\ 1 & x, y \in E and y < x, \\ 4 & x, y \in L ∖ E and x \leq y, \\ 1 & x \in L ∖ E, y \in E and y < x, \\ 6 & x \in L ∖ E, y \in E and x \leq y, \\ 8 & x \in E, y \in L ∖ E and x \leq y, \\ 1 & x \in E, y \in L ∖ E and y < x, \\ 1 & x, y \in L ∖ E and y < x . \end{matrix}$ Then the triple (L, → , E) is a weak implicative groupoid but it is not a weak extended implicative groupoid.

(iv) Let L = {a, b, c, d, e}, E = {d, e} and define the operation → on L as follows:

→	a	b	c	d	e
a	d	d	a	d	d
b	a	d	d	d	d
c	a	a	d	d	d
d	a	a	a	d	d
e	a	a	a	a	d

Then it is easy to see that (wei1) - (wei3) holds and → is transitive on E. Thus (pwei) holds. Since a → b, b → c ∈ E and a → c ∉ E, it follows that (L, → , E) is a pre-weak extended implicative groupoid but it is not a weak extended implicative groupoid.

(v) Let L = {a, b, c}, E = {b, c} and the operation → on L is defined as follows:

→	a	b	c
a	c	c	c
b	a	b	b
c	a	a	c

It is clear that b → c ∈ E, such that

$\begin{matrix} (a \to b) \to (a \to c) \in E, (b \to b) \to (b \to c) \in E, \\ (c \to b) \to (c \to c) \in E \end{matrix}$

and

$\begin{matrix} (c \to a) \to (b \to a) \in E, (c \to b) \to (b \to b) \in E, \\ (c \to c) \to (b \to c) \notin E . \end{matrix}$

Hence (L, → , E) is a right extended implicative groupoid but it is not a left extended implicative groupoid.

(vi) Let L = {a, b, c}, E = {b, c} and the operation → on L is defined as follows:

→	a	b	c
a	c	c	c
b	a	c	b
c	a	a	b

Then (L, → , E) is a left extended implicative groupoid but it is not a right extended implicative groupoid, since b → c ∈ E but (b → b) → (b → c) = a ∉ E.

(vii) Let (L, → , E) be the weak extended implicative groupoid as introduced in (vi). Then it is easy to see that the order induced by → is a linear order and since L is a finite set, we get L is a complete lattice. It follows that (L, → , E) is a complete weak extended implicative groupoid.

(viii) Define the operation → on L = {a, b, c, d} as follows:

→	a	b	c	d
a	c	c	c	d
b	a	d	d	d
c	a	b	c	d
d	a	b	a	c

Then by considering the upper set E = {c, d}, we get that (L, → , E) is a compatible pre-weak implicative groupoid.

(ix) Define the operation → on L = {a, b, c, d, e} as follows:

→	a	b	c	d	e
a	e	e	c	e	e
b	a	e	e	e	e
c	a	a	e	e	a
d	a	a	a	e	a
e	a	a	a	a	e

Then by considering the upper set E = {c, d, e}, we get that (L, → , E) is a weak extended implicative groupoid. Since a → b ∈ E, and $(a \to c) \to (b \to c) = c \to e = a \notin E,$ it follows that (L, → , E) is not a compatible pre-weak implicative groupoid.

(x) Let L = {a, b, c, d}, E = {c, d}. Define the operation → on L as follows:

→	a	b	c	d
a	d	d	d	d
b	a	d	d	d
c	a	b	c	d
d	a	a	a	d

Then (L, → , E) is a (left or right) distributive extended implicative groupoid.

(xi) Define the operation → on L = {a, b, c, d, e} as follows:

→	a	b	c	d	e
a	c	e	c	d	e
b	a	e	e	c	e
c	a	b	e	e	e
d	a	b	a	e	e
e	a	a	a	a	e

Then by considering the complete distributive lattice E = {c, d, e}, we get that the triple (L, → , E) is a weak extended implicative groupoid. Let A = {b}, A′ = {b}, B = {c} and B′ = {c, d}. Then $\begin{matrix} A \to B = {b \to c} = e and \\ A^{'} \to B^{'} = {b \to c, b \to d} = {c, e} . \end{matrix}$ As $\begin{matrix} U (A) = U (A^{'}) = {b, c, d, e} and \\ L (B) = L (B^{'}) = {a, b, c}, \end{matrix}$ while

$L (A \to B) = {a, b, c, d, e} \neq {a, b, c} = L (A^{'} \to B^{'}),$

we conclude that (L, → , E) is not a distributive extended implicative groupoid.

Remark 3.15 By Proposition 3.8, we can see that the condition (cpwe) implies (wei4). But as in Example 3.14(ix), the converse is not true, in general.

4 MacNeille completion of (weak) extended implicative groupoids

In this section, the MacNeille completion of (weak) extended implicative groupoids and its properties are considered.

Let (L, → , E) be a weak extended implicative groupoid and ≤ denotes the induced partial order on L. For $A, A^{'} \in P (L)$ , define a binary relation ≃ on L by A ≃ A′ if and only if $L (A) = L (A^{'})$ . It is easy to see that the relation ≃ is an equivalence relation on $P (L)$ and ≃ is a congruence on $P (L)$ if L is a lower set preserving. Let $K_{≃} : = {[A] : A \in P (L)}$ and for any [A] , [B] ∈ K_≃, define [A] ≤ _{k
_≃} [B] if and only if $L (A) \subseteq L (B)$ . Then ≤_{K
_≃} is a partial order on K_≃. Let $E = {[Z] : Z \subseteq E} \subseteq K_{≃} .$ Further, define the operation →_{K
_≃} on K_≃, such that for any [A] , [B] ∈ K_≃, we have $[A] \to_{K} [B] = [L (A) \to U (L (B))] .$ Then the following results hold.

Theorem 4.1 Let (L, → , E) be a weak extended implicative groupoid with (lpei). Then the following statements are equivalent:

(i) (K_≃, ≤ _{K
_≃}) is a complete lattice.

(ii) $(E, \leq_{K_{≃}})$ is a complete lattice.

(iii) The poset {[e] : e ∈ E} has the greatest element [⊤], where ⊤ ∈ E.

(iv) E possesses the greatest element ⊤ with respect to the order induced by →.

Proof. (iii) ⇔ (iv) If (iii) holds, then there exists ⊤ ∈ E such that for any e ∈ E, we have [e] ≤ _{K
_≃} [⊤]. But this holds precisely when ↓e⊆ ↓ ⊤, for any e ∈ E. Especially, we have e∈ ↓ ⊤, for any e ∈ E, which implies e≤ ⊤. Hence ⊤ is the greatest element of E. Conversely, assume [⊤] belongs to the poset {[e] : e ∈ E}. For any e ∈ E, we have ↓e⊆ ↓ ⊤, which implies [e] ≤ _{K
_≃} [⊤]. Consequently, [⊤] is the greatest element of the poset {[e] : e ∈ E}.

(i) ⇔ (iii) First, assume (K_≃, ≤ _{K
_≃}) is a complete lattice. Since {[e] : e ∈ E} ⊆ K_≃, the poset {[e] : e ∈ E} attains its least upper bound. Let [A] = ⋁ {[e] : e ∈ E}, for some A ⊆ E. We claim that A is a singleton. If α, β ∈ A such that α ≠ β, then [A] ^< ≠ _{K
_≃} [α], which is a contradiction. Hence there exists ⊤ ∈ E such that [⊤] = ⋁ {[e] : e ∈ E}. Conversely, assume [⊤] = ⋁ {[e] : e ∈ E} ∈ K_≃, where ⊤ ∈ E. Obviously, [⊤] is an upper bound and [{∅}] is a lower bound, for any B ⊆ K_≃. Consider an arbitrary subset U of K_≃. Let U = {[U_i] : i ∈ I}, for some $U_{i} \in P (L)$ , i ∈ I. Let U₀ = ⋃ _i∈IU_i and U₁ = ⋂ _i∈IU_i. Then [U₀] , [U₁] ∈ K_≃, [U₀] is the least upper bound of U and [U₁] is the greatest lower bound of U. Consequently, (K_≃, ≤ _{K
_≃}) is a complete lattice.

(ii) ⇔ (iii) If $(E, \leq_{K_{≃}})$ is a complete lattice, since ${[e] : e \in E} \subseteq E$ , then the poset {[e] : e ∈ E} attains its greatest upper bound, which is indeed a singleton. Conversely, assume [⊤] = ⋁ {[e] : e ∈ E} ∈ K_≃, where ⊤ ∈ E. Obviously, [⊤] is an upper bound and [{∅}] is a lower bound, for any $B \subseteq E$ . Consider an arbitrary subset V of $E$ . Let V = {[V_j] : j ∈ J}, for some $V_{j} \in P (L)$ , j ∈ J. Let V₀ = ⋃ _j∈JU_j and V₁ = ⋂ _j∈JV_j. Then $[V_{0}], [V_{1}] \in E$ , [V₀] is the least upper bound of V, and [V₁] is the greatest lower bound of V. Consequently, $(E, \leq_{K_{≃}})$ is a complete lattice.

□

Example 4.2 Define the operation → on L = {a, b, c, d, e} as follows:

→	a	b	c	d	e
a	c	a	c	c	c
b	a	c	c	c	c
c	a	a	c	c	c
d	a	a	a	c	a
e	a	a	a	a	c

Then by considering E = {c, d, e}, we get that the triple (L, → , E) is a weak extended implicative groupoid. It is easy to see that K_≃ = {⊥ _{K
_≃}, [a] , [b] , [c] , [d] , [e]}, where ⊥_{K
_≃} = [L], $L (a) = {a}$ , $L (b) = {b}$ , $L (c) = {a, b, c}$ , $L (d) = {a, b, c, d}$ and $L (e) = {a, b, c, e}$ . Neither of the lower sets ↓d nor ↓e contains the other one, hence (K_≃, ≤ _{K
_≃}) is not a (complete) lattice. Thus the weak extended implicative groupoid (L, → , E) does not possess a MacNeille completion. This was anticipated by Theorem 4.1, since the poset E does not admit the greatest element.

It the following, we see that unlike weak extended order algebras, a weak extended implicative groupoid does not always admit a MacNeille completion by Theorem 4.1 and Example 4.2. On the other hand, if this completion exists, then it is a complete weak extended implicative groupoid.

Proposition 4.3 The following statements hold:

(i) If (L, → , E) is a weak extended implicative groupoid with (lpei), then →_{K
_≃} is an extension of ≤_{K
_≃}, i.e., for all μ, ν ∈ K_≃ $μ \to_{K_{≃}} ν \in E if and only if μ \leq_{K_{≃}} ν .$ (ii) If a weak extended implicative groupoid (L, → , E) admits a MacNeille completion (K_≃, ≤ _{K
_≃}), then $(K_{≃}, \to_{K_{≃}}, E)$ is a complete weak extended implicative groupoid.

(iii) If (L, → , E) is an extended implicative groupoid with (lpei), then →_{K
_≃} is an extension of →, i.e., for all a, b ∈ L $[a] \to_{K_{≃}} [b] = [a \to b] .$ (iv) If (L, → , E) is a distributive extended implicative groupoid with (lpei), then for all α = [A] ∈ K_≃ and β = [B] ∈ K_≃, $α \to_{K_{≃}} β = [L (A) \to B] .$

Proof. (i) Assume μ = [C] and ν = [D], for some C, D ⊆ L. Then $μ \to_{K_{≃}} ν \in E$ , if and only if $[L (C) \to U (L (D))] \in E$ , or equivalently $L (C) \to U (L (D)) \in E$ , which holds precisely when $L (C) \subseteq L (U (L (D))) = L (D)$ . Thus μ ≤ _{K
_≃}ν. Consequently →_{K
_≃} is an extension of ≤_{K
_≃}. (ii) Assume (K_≃, ≤ _{K
_≃}) is the MacNeille completion of (L, → , E). From Theorem 4.1, it follows that the induced order by →_{K
_≃} on K coincides with ≤_{K
_≃}. Consequently $(K_{≃}, \to_{K_{≃}}, E)$ is a complete weak extended implicative groupoid. (iii) Since {a → b} ⊆ (↓ a) → (↑ b), it easily follows that $[a] \to_{K_{≃}} [b] = [(↓ a) \to (↑ b)] \leq_{K_{≃}} [a \to b] .$ Let c → d ∈ (↓ a) → (↑ b). Then c ≤ a and b ≤ d. Since b → d ∈ E, by (rei), we have $(c \to b) \to (c \to d) \in E .$ Similarly, since c → a ∈ E, by (lei), we get that (a → b) → (c → b) ∈ E . By transitivity condition, we have (a → b) → (c → d) ∈ E . Hence $(↓ a) \to (↑ b) \subseteq U (a \to b)$ , and so $L (a \to b) = L (U (a \to b)) \subseteq L ((↓ a) \to (↑ b)),$ which proves the converse inequality in (iii).

(iv) Since $L (U (L (B))) = L (B)$ , we have $L (L (A) \to U (L (B))) = L (L (A) \to B),$ which is the desired result.□

The next example shows that a weak extended implicative groupoid does not satisfy the assertion (iii) of the previous proposition, even if it admits a MacNeille completion.

Example 4.4 Define the operation → on L = {a, b, c, d, e} as follows:

→	a	b	c	d	e
a	c	e	c	c	c
b	b	e	e	c	c
c	a	a	c	c	c
d	a	a	a	c	c
e	a	a	a	a	c

Then by considering E = {c, d, e}, the triple (L, → , E) is a weak extended implicative groupoid. It is easy to see that K_≃ = {⊥ _{K
_≃}, [a] , [b] , [c] , [d] , [e]}, where ⊥_{K
_≃} = [L], $L (a) = {a}$ , $L (b) = {a, b}$ , $L (c) = {a, b, c}$ , $L (d) = {a, b, c, d}$ and $L (e) = L$ . Obviously, (K_≃, ≤ _{K
_≃}) is a complete lattice. Hence the weak extended implicative groupoid (L, → , E) possesses a MacNeille completion. Since c → d ∈ E and (b → c) → (b → d) = (e → c) = a ∉ E, then (L, → , E) does not satisfy (rei). Hence (L, → , E) is not an extended implicative groupoid. On the other hand, $[a] \to_{K_{≃}} [b] = [↓ a \to U (↓ b)] = [{c, e}] = [c],$ while [a → b] = [e] . Therefore, →_{K
_≃} is not an extension of ≤_{K
_≃}.

Definition 4.5 Let (L, → , E) be a weak extended implicative groupoid. The triple (L, → , E) is called:

(i) Idempotent, if for all a, b ∈ L, it satisfies in the following condition:

(idwei) a → b ∈ E if and only if a → (a → b) ∈ E.

(ii) Commutative, if for all a, b, c ∈ L, it satisfies in the following condition:

(cmwei) a → (b → c) ∈ E if and only if b → (a → c) ∈ E.

(iii) Associative, if for all A, B, C ⊆ L, it satisfies in the following condition:

(asei) $L (L (A) \to (L (B) \to C)) = L (L ({x : L (A) \subseteq L (L (B) \to x)}) \to C)$ .

By a similar discussion like [13, Lemma 37], we get that a complete distributive extended implicative groupoid (L, → , E) is associative if and only if for all a, b, c ∈ L $a \to (b \to c) = (\land {x : a \to (b \to x) \in E}) \to c .$

Proposition 4.6 Let L = (L, → , E) be an extended implicative groupoid with (lpei) such that E has the greatest upper bound ⊤ ∈ E. Then, the following statements hold:

(i) $(K_{≃}, \to_{K_{≃}}, E)$ is a complete extended implicative groupoid.

(ii) If L is distributive, then $(K_{≃}, \to_{K_{≃}}, E)$ is distributive.

(iii) If L is compatible, then $(K_{≃}, \to_{K_{≃}}, E)$ is compatible.

(iv) If L is left distributive, then $(K_{≃}, \to_{K_{≃}}, E)$ is left distributive.

(v) If L is right distributive, then $(K_{≃}, \to_{K_{≃}}, E)$ is right distributive.

(vi) If L is a lower set preserving, then $(K_{≃}, \to_{K_{≃}}, E)$ is a lower set preserving.

(vii) If L is an upper set preserving, then $(K_{≃}, \to_{K_{≃}}, E)$ is an upper set preserving.

(viii) L is idempotent if and only if $(K_{≃}, \to_{K_{≃}}, E)$ is idempotent.

(ix) If L is commutative, then $(K_{≃}, \to_{K_{≃}}, E)$ is commutative.

(x) If L is distributive and associative, then $(K_{≃}, \to_{K_{≃}}, E)$ is associative.

Proof. The proofs of (i) and (ii) are quite similar to the proof of [13, Proposition 29].

(iii) Let $K ∖ E = {[U] : \emptyset \neq U \subseteq L ∖ E} .$ We need to prove that, for any $[A], [B] \in K ∖ E$ , $\begin{matrix} [A] \to_{K} [B] \in E if and only if \\ ([A] \to_{K_{≃}} [C]) \to_{K_{≃}} ([B] \to_{K_{≃}} [C]) \in E, \end{matrix}$ for any $[C] \in E$ . For this, let ∅ ≠ C ⊆ E. Since (L, → , E) is an extended implicative groupoid, by Proposition 4.3(iii), we have $\begin{matrix} [A] \to_{K_{≃}} [B] \in E if and only if [A \to B] \in E \\ if and only if A \to B \in E . \end{matrix}$ Moreover, since (L, → , E) is compatible, we have (A → C) → (B → C) ∈ E, which is equivalent to $[(A \to C) \to (B \to C)] \in E$ . By Proposition 4.3, we have $([A] \to_{K_{≃}} [C]) \to_{K_{≃}} ([B] \to_{K_{≃}} [C]) \in E,$ which completes the proof. (iv) Let L be left distributive and A, B ∈ K_≃. By Proposition 3.11(ii), it is enough to show that (∨ _{K
_≃}A) → _{K
_≃}B = ∧ _{K
_≃} (A → _{K
_≃}B). Let A = {[U] : U ∈ A^*} and B = {[V] : V ∈ B^*} where A^*, B^* are subsets of L. For this, the following is obtained based on Proposition 4.3 and the fact that L is left distributive: $\begin{matrix} (\lor_{K_{≃}} A) \to_{K_{≃}} B \\ = {[U (\cup {L (U) | [U] \in A})] \to_{K_{≃}} [V] | [V] \in B} \\ = {[U (\cup {L (U) | [U] \in A^{*}}) \to V] : V \in B^{*}} \\ = {[\lor A^{*} \to V] : V \in B^{*}} \\ = {[\land (A^{*} \to V)] : V \in B^{*}} \\ = {[\cup (U \to V) : U \in A^{*}] : V \in B^{*}} \\ = {\land_{K_{≃}} [U \to V] : U \in A^{*}, V \in B^{*}} \\ = {\land_{K_{≃}} ([U] \to_{K_{≃}} [V]) : U \in A^{*}, V \in B^{*}} \\ = \land_{K_{≃}} (A \to_{K_{≃}} B) . \end{matrix}$ Hence (iv) holds. Accordingly, (v) holds.

(vi) Let L be a lower set preserving and consider A, A′, B, B′ ∈ K_≃ such that $L_{K_{≃}} (A) = L_{K_{≃}} (A^{'})$ and $L_{K_{≃}} (B) = L_{K_{≃}} (B^{'})$ . Assume that $\begin{matrix} A = {[α] : α \in A^{*}}, A^{'} = {[α^{'}] : α^{'} \in C^{*}}, \\ B = {[β] : β \in B^{*}}, B^{'} = {[β^{'}] : β^{'} \in D^{*}}, \end{matrix}$ for $A^{*}, B^{*}, C^{*}, D^{*} \in P (L)$ . Since $L_{K_{≃}} (A) = L_{K_{≃}} (A^{'})$ , we get that $\begin{matrix} ↓_{k_{≃}} [\cup {α : α \in A^{*}}] = ↓_{k_{≃}} [\cup {α^{'} : α^{'} \in C^{*}}] . \end{matrix}$ Then $L (\cup {α : α \in A^{*}}) = L (\cup {α^{'} : α^{'} \in C^{*}}) .$ Similarly, since $L_{K_{≃}} (B) = L_{K_{≃}} (B^{'})$ , we have $L (\cup {β : β \in B^{*}}) = L (\cup {β^{'} : β^{'} \in D^{*}}) .$ In addition, since L is a lower set preserving, we have $\begin{matrix} L (\cup {α \to β : α \in A^{*}, β \in B^{*}}) \\ = L (\cup {α^{'} \to β^{'} : α^{'} \in C^{*}, β^{'} \in D^{*}}) . \end{matrix}$ Hence by Proposition 4.3 and [13, Lemma 23(1)], we obtain: $\begin{matrix} L_{K_{≃}} (A \to_{K_{≃}} B) \\ = L_{K_{≃}} (\cup {[α] \to_{K_{≃}} [β] : α \in A^{*}, β \in B^{*}}) \\ = L_{K_{≃}} (\cup {[α \to β] : α \in A^{*}, β \in B^{*}}) \\ = L_{K_{≃}} ([\cup {α \to β : α \in A^{*}, β \in B^{*}}]) \\ = L_{K_{≃}} ([L (\cup {α \to β : α \in A^{*}, β \in B^{*}})]) \\ = L_{K_{≃}} ([L (\cup {α^{'} \to β^{'} : α^{'} \in C^{*}, β^{'} \in D^{*}})]) \\ = L_{K_{≃}} (\cup {[α^{'}] \to_{K_{≃}} [β^{'}] : α^{'} \in C^{*}, β^{'} \in D^{*}}) \\ = L_{K_{≃}} (A^{'} \to_{K_{≃}} B^{'}), \end{matrix}$ The proof of (vii) is exactly similar to the proof of (iii). The proof of (viii), (ix) and (x) are similar to [8, Proposition 4.3], [13, Proposition 34] and [13, Proposition 38], respectively.□

We show that the converse of some statements in Proposition 4.6, does not hold.

Example 4.7 (i) Let L = {a, b, c, d, e}, E = {c, d, e} and the operation → on L is defined as follows:

→	a	b	c	d	e
a	c	c	c	c	c
b	a	c	e	c	c
c	a	a	c	c	c
d	a	a	a	c	c
e	a	a	a	a	c

Then (L, → , E) is a (weak) extended implicative groupoid. Now,

(a) (L, → , E) is not left distributive, because $↑ b = U ({a, b}) = U ({b})$ but $\begin{matrix} ↓ e = L (b \to c) \neq L ({a \to c, b \to c}) \\ = L ({c, e}) = ↓ c . \end{matrix}$ (b) (L, → , E) is not right distributive, because $↓ c = L ({c}) = L ({c, e})$ but $↓ e = L (b \to c) \neq L ({b \to c, b \to e}) = ↓ c .$ (c) (L, → , E) is not distributive, because $↑ b = U ({a, b}) = U ({b})$ and $L ({c}) = L ({c})$ but

$↓ e = L (b \to c) \neq L ({a \to c, b \to c}) = L ({c, e}) = ↓ c .$

(d) (L, → , E) is not commutative, because e → (b → c) = c ∈ E but b → (e → c) = a ∉ E .

(e) (L, → , E) is not lower set preserving, because $L ({b}) = L ({b, c})$ and $L ({c}) = L ({c})$ but $↓ e = L ({b \to c}) \neq L ({b \to c, c \to c}) = ↓ c .$ (f) (L, → , E) is not upper set preserving, because $U ({c}) = U ({b, c})$ and $U ({c}) = U ({c})$ but $\begin{matrix} ↑ c = U ({c \to c}) \neq U ({b \to c, c \to c}) \\ = U ({e, c}) = ↑ e = {e} . \end{matrix}$ It is easy to see that (L, → , E) admits MacNeil completion $(K_{≃}, \to_{K_{≃}}, E)$ , where $K_{≃} = {[a], [b], [c], [d], [e]}, E = {[c], [d], [e]}$ and the operation →_{K
_≃} on K_≃ is defined as follows:

→_{K _≃}	[a]	[b]	[c]	[d]	[e]
[a]	[c]	[c]	[c]	[c]	[c]
[b]	[a]	[c]	[c]	[c]	[c]
[c]	[a]	[a]	[c]	[c]	[c]
[d]	[a]	[a]	[a]	[c]	[c]
[e]	[a]	[a]	[a]	[a]	[c]

It is easy to see that $(K_{≃}, \to_{K_{≃}}, E)$ is a (left-right) distributive commutative extended implicative groupoid that preserves both lower and upper sets. Hence the converse of statements (ii) , (iv) , (v) , (vi) , (vii) , (ix) in Proposition 4.6, does not hold.

(ii) Let L = {a, b, c, d, e}, E = {c, d, e} and the operation →¹ on L is defined as follows:

→¹	a	b	c	d	e
a	c	c	e	c	c
b	a	c	c	c	c
c	a	a	c	c	c
d	a	a	a	c	c
e	a	a	a	a	c .

Then (L, → ¹, E) is a (weak) extended implicative groupoid, but it is not compatible, since a → ¹b ∈ E but $(a \to^{1} c) \to^{1} (b \to^{1} c) = e \to^{1} c = a \notin E .$ It is easy to see that (L, → ¹, E) admits MacNeil completion $(K_{≃}, \to_{K_{≃}}^{1}, E)$ , where $K_{≃} = {[a], [b], [c], [d], [e]}, E = {[c], [d], [e]}$ and the operation $\to_{K_{≃}}^{1}$ on K_≃ is defined as follows

$\to_{K_{≃}}^{1}$	[a]	[b]	[c]	[d]	[e]
[a]	[c]	[c]	[c]	[c]	[c]
[b]	[a]	[c]	[c]	[c]	[c]
[c]	[a]	[a]	[c]	[c]	[c]
[d]	[a]	[a]	[a]	[c]	[c]
[e]	[a]	[a]	[a]	[a]	[c]

Then $(K_{≃}, \to_{K_{≃}}^{1}, E)$ is a compatible extended implicative groupoid. Hence the converse of statement (iii) in Proposition 4.6, does not hold.

Based on the above results, the MacNeille completion provides us to obtain the completeness condition of an extended implicative groupoid. Furthermore, the properties of an extended implicative groupoid are satisfied by its MacNeille completion, as shown in [13].

5 The relation between extended implicative groupoids and residuated lattices

In this section, we briefly discuss the adjoint product and its properties on complete distributive extended implicative groupoids. Then the relation between symmetrical complete extended implicative groupoids and complete residuated lattices is evaluated.

Let (L, → , E) be a complete distributive extended implicative groupoid. For any a, b ∈ L, we define their adjoint product by setting a ⊗ b = ∧ {x ∈ L : b ≤ a → x}. The properties of ⊗ are summarized in the following proposition.

Proposition 5.1 Let (L, → , E) be a complete distributive extended implicative groupoid. Then the following statements hold for any a, b, c ∈ L:

(i) If (E, ⪯) attains its greatest lower bound ⊥_E ∈ E, then a ⊗ ⊥ _E = a.

(ii) If a ≤ b, then a ⊗ c ≤ b ⊗ c.

(iii) a ⊗ (a → b) ≤ b ≤ a → (a ⊗ b).

(iv) For all x ∈ L, $\begin{matrix} ⊤ \otimes a = a if and only if ⊤ \to x \geq a \\ if and only if x \geq a . \end{matrix}$ (v) (L, → , E) is commutative if and only if its adjoint product ⊗ is commutative, i.e. a ⊗ b = b ⊗ a, for all a, b ∈ L.

(vi) (L, → , E) is associative if and only if ⊗ is associative, i.e. (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c), for all a, b, c ∈ L.

(vii) ⊗ and → form an adjoint pair, i.e. for all x, y, z ∈ L $x \otimes y \leq z if and only if y \leq x \to z .$

Proof. The proofs of (i) - (vi) are similar to [13, Propositions 31, 35, 40]. The proof of (vii) is straightforward. □

Now, we first recall the notion of a residuated lattice.

Definition 5.2. [2] A residuated lattice-order monoid (or residuated lattice) is an algebraic structure (L, ∧ , ∨ , ⊗ , → , ⇝ , e) of type (2, 2, 2, 2, 2, 0) such that:

(rl₁) (L, ∧ , ∨) is a lattice,

(rl₂) (L, ⊗ , e) is a monoid,

(rl₃) the operations → and ⇝ are the residuals of ⊗, in fact, for all x, y, z ∈ L $\begin{matrix} y \leq x \to z if and only if x \otimes y \leq z \\ if and only if x \leq y ⇝ z . \end{matrix}$ In the residuated lattice L, if the operation ⊗ is commutative, then L is called a commutative residuated lattice. In this case, it is easy to see that the operations → and ⇝ coincide with each other.

As it was already mentioned in [8, 13], Proposition 5.1 determines the important differences between complete distributive extended implicative groupoids and complete residuated lattices. To obtain the close relationship between complete distributive extended implicative groupoids and complete residuated lattices, we need an additional condition which is discussed in the following.

Definition 5.3 A weak extended implicative groupoid (L, → , E) is called symmetrical, if there exists another weak extended implicative groupoid (called its dual) (L, ⇝ , E), such that the implications → and ⇝ form a Galois pair [→ , ⇝], it means that → and ⇝ induce the same order ≤ on L and for all b, x, y ∈ L, y ≤ x ⇝ b if and only if x ≤ y → b.

Some properties of symmetrical weak extended implicative groupoids are provided in the following lemma.

Lemma 5.4 Let (L, → , E) be a symmetrical weak extended implicative groupoid with dual (L, ⇝ , E). Then the following statements hold:

(i) For any e ∈ E and a, x ∈ L, $\begin{matrix} a \leq e \to x if and only if a \leq x \\ if and only if a \leq e ⇝ x . \end{matrix}$ (ii) If (L, → , E) or (L, ⇝ , E) is a right weak extended implicative groupoid, then both of them are extended implicative groupoids.

Proof. The proof is similar to the proof of [8, Proposition 5.3]. □

The following proposition plays a crucial role in the main result of this paper, which provides a specific description of the property of symmetrical complete distributive extended implicative groupoid.

Proposition 5.5 Let (L, → , E) be a symmetrical complete distributive extended implicative groupoid with the adjoint product ⊗ and dual (L, ⇝ , E). If (E, ⪯) attains its greatest lower bound ⊥_E ∈ E, then ⊥_E ⊗ b = b ⊗ ⊥ _E = b for all b ∈ L.

Proof. Suppose that (E, ⪯) attains its greatest lower bound ⊥_E ∈ E and b ∈ L. Since $⊥_{E} \otimes b = \land {x \in L : b \leq ⊥_{E} \to x},$ for any x ∈ L, by Lemma 5.4(i), b ≤ ⊥ _E → x if and only if b ≤ x. Since b ≤ b, we have b ≤ ⊥ _E → b, and so ⊥_E ⊗ b = b. Now, by definition of ⊗, we have $b \otimes ⊥_{E} = \land {x \in L : ⊥_{E} \leq b \to x} .$ If x ∈ L such that ⊥_E ≤ b → x, then b ≤ x. Moreover, since ⊥_E ∈ E is the greatest lower bound of E, we have ⊥_E ≤ b → b. Hence b ⊗ ⊥ _E = b. Thus ⊥_E ⊗ b = b ⊗ ⊥ _E = b. □

The following theorem is the main result of this article, which describes the relation between symmetrical complete distributive extended implicative groupoids and complete residuated lattices.

Theorem 5.6 Let (L, → , E) be a complete associative distributive extended implicative groupoid with the adjoint product ⊗ such that (E, ⪯) attains its greatest lower bound ⊥_E ∈ E. Then the following statements hold:

If (L, → , E) is symmetrical with dual (L, ⇝ , E), then (L, ∧ , ∨ , ⊗ , → , ⇝ , ⊥ _E) is a complete residuated lattice.

If (L, ⊗) is commutative, then (L, ∧ , ∨ , ⊗ , → , ⊥ _E) is a commutative complete residuated lattice.

Proof. (i) By assumption, (L, ≤) is a complete lattice, where ≤ is the order induced by ⇝ or →. Now, considering (L, ⊗ , ⊥ _E). Then by Propositions 5.1(vi) and 5.5, we have (L, ⊗ , ⊥ _E) is a monoid. Furthermore, by Proposition 5.1(vii), the operations → and ⊗ form an adjoint pair. In addition, the operations → and ⇝ are Galois pairs. Hence considering these two facts, it is concluded that we have the following residuation properties, for any x, y, z ∈ L, $\begin{matrix} y \leq (x \to z) if and only if x \otimes y \leq z \\ if and only if x \leq (y ⇝ z) . \end{matrix}$ Therefore, (L, ∧ , ∨ , ⊗ , → , ⇝ , ⊥ _E) is a complete residuated lattice.

(ii) By assumption, (L, ≤) is a complete lattice, where ≤ is the order induced by →. Since (L, ⊗) is commutative, by Proposition 5.1(i) and (vi), it follows that (L, ⊗ , ⊥ _E) is a commutative monoid. From Proposition 5.1(vii), the operations → and ⊗ form an adjoint pair. Hence, for all x, y, z ∈ L, $x \leq (y \to z) if and only if y \otimes x = x \otimes y \leq z .$ Therefore, (L, ∧ , ∨ , ⊗ , → , ⊥ _E) is a commutative complete residuated lattice.□

The above theorem naturally arises a few questions in mind that we discuss them in the following remark and Theorem 5.8.

Remark 5.7 In Theorem 5.6, we assumed that the underlying extended implicative groupoid (L, → , E) is associative, complete, distributive, and either commutative or symmetrical. Now, we explain the reason for considering these conditions and why they are necessary. First, it is worth noting that the existence of operation ⊗ requires (L, → , E) to be complete and distributive. The associativity assumption is a fundamental criterion by (rl₂). Finally, based on the proof of Theorem 5.6, either the commutativity or the symmetry assumption is necessary for (L, ⊗ , ⊥ _E) so that it can be considered as a monoid.

Now, the main question raised is whether the converse of Theorem 5.6 holds or not? The following theorem confirms the positive answer to the question.

Theorem 5.8

Let (L, ∧ , ∨ , ⊗ , → , e) be a commutative complete residuated lattice and E = {λ : λ = x → y, x ≤ y, x, y, λ ∈ L} such that E be an upper bound of L ∖ E respect to ≤. Then (L, → , ⊗ , E) is a commutative associative complete distributive extended implicative groupoid such that e = ⊥ _E ∈ E.

Let (L, ∧ , ∨ , ⊗ , → , ⇝ , e) be a complete residuated lattice and E = {λ : λ = x → y, x ≤ y, x, y, λ ∈ L} such that E be an upper bound of L ∖ E with respect to ≤. Then (L, → , ⇝ , ⊗ , E) is a symmetrical associative complete distributive extended implicative groupoid such that e = ⊥ _E ∈ E.

Proof. (i) Since (L, ∧ , ∨ , ⊗ , → , e) is a commutative complete residuated lattice, (L, ⊗) is commutative. We show that (L, → , ⊗ , e) is a commutative associative complete distributive extended implicative groupoid such that e = ⊥ _E ∈ E. By assumption, for any a ∈ L ∖ E and b ∈ E, we have a ≤ b. Thus a → b ∈ E. Consequently, (wei1) holds. Since for any a ∈ L, we have a → a ∈ E, then (wei2) holds. If a → b, b → a ∈ E for all a, b ∈ L, then a ≤ b and b ≤ a, which implies a = b, and so (wei3) holds. Finally, for all a, b, c ∈ L, we have a ≤ b ≤ c. Now, if a → b, b → c ∈ E, then a ≤ c. Hence (wei4) holds, and consequently (L, → , E) is a weak extended implicative groupoid. Let a, b ∈ L such that a ≤ b and c ∈ L. Based on the properties of the commutative residuated lattice, which are similar to Proposition 5.1(ii) and (iii), we conclude that $a \otimes (b \to c) \leq b \otimes (b \to c) \leq c .$ Hence a ⊗ (b → c) ≤ c. However, from (rl₃), we have b → c ≤ a → c. In other words, (L, → , E) is a left extended implicative groupoid. Furthermore, let a, b ∈ L such that a ≤ b and c ∈ L. By (rl₃), since c → a ≤ c → a, we have c ⊗ (c → a) ≤ a, and so c ⊗ (c → a) ≤ b. Once again from (rl₃), it follows that c → a ≤ c → b. Therefore, (rei) holds. Now, since (L, → , E) is a right and a left extended implicative groupoid, by Definition 3.1, (L, → , E) is an extended implicative groupoid. In the sequel, by assumption and Definition 3.6, (L, → , E) is a complete extended implicative groupoid. Consider arbitrary subsets A, B, A′, B′ of L satisfying $U (A) = U (A^{'})$ and $L (B) = L (B^{'})$ . Let $z \in L (A \to B)$ . Then for some a ∈ A and b ∈ B, we have z ≤ a → b. From z ≤ a → b, we have a ⊗ z ≤ b, which implies $a \otimes z \in L (B) = L (B^{'})$ . Hence there exists b′ ∈ B′ such that a ⊗ z ≤ b′. From commutativity of the adjoint product, z ⊗ a ≤ b′, or equivalently a ≤ z → b′. It implies $z \to b^{'} \in U (A) = U (A^{'})$ . Hence there exists a′ ∈ A′ such that a′ ≤ z → b′. From this, we get that z ⊗ a′ = a′ ⊗ z ≤ b′ or equivalently z ≤ a′ → b′. Hence we prove that $L (A \to B) \subseteq L (A^{'} \to B^{'})$ . By a similar discussion, we can show that $L (A \to B) \supseteq L (A^{'} \to B^{'})$ . Thus $L (A \to B) = L (A^{'} \to B^{'})$ and consequently (L, → , E) forms a distributive complete extended implicative groupoid. For any r ∈ E, since (L, ⊗ , e) is a commutative monoid, we have e ⊗ r = r ⊗ e = r. Hence for any r ∈ E, r ≤ e → r, and so for any r ∈ E, e ≤ r or equivalently e = ⊥ _E. Finally, by assumption and Proposition 5.1(v) and (vi), we get that (L, → , ⊗ , E) is a commutative associative complete distributive extended implicative groupoid.

(ii) (L, →) is symmetrical with dual (L, ⇝), since (L, ∧ , ∨ , ⊗ , → , ⇝ , e) is a complete residuated lattice. We show that (L, → , ⊗ , E) is a complete associative distributive extended implicative groupoid such that e = ⊥ _E ∈ E. Similar to the proof of (i), we get that (L, → , E) is a weak extended implicative groupoid. Let a, b ∈ L such that a ≤ b and c ∈ L. Since c → a ≤ c → a, by (rl₃), we get that c ⊗ (c → a) ≤ a, and so c ⊗ (c → a) ≤ b. Hence by (rl₃), c ≤ (c → a) ⇝ b. Since [→ , ⇝] form a Galois pair, we have c → a ≤ c → b. In other words, (L, → , E) is a right extended implicative groupoid. Now, by Lemma 5.4(ii), (L, → , E) is an extended implicative groupoid. Consider arbitrary subsets A, B, A′, B′ of L satisfying $U (A) = U (A^{'})$ and $L (B) = L (B^{'})$ . Let $z \in L (A \to B)$ . Then for some a ∈ A and b ∈ B, z ≤ a → b. Since z ≤ a → b, we get that a ⊗ z ≤ b, which implies $a \otimes z \in L (B)$ . Hence there exists b′ ∈ B′ such that a ⊗ z ≤ b′, and so z ≤ a → b′. By the symmetric property it follows that a ≤ z ⇝ b′. This implies $z ⇝ b^{'} \in U (A)$ . Hence there exists a′ ∈ A′ such that a′ ≤ z ⇝ b′. Once again, using the symmetric property, we get that z ≤ a′ → b′. Hence we proved $L (A \to B) \subseteq L (A^{'} \to B^{'})$ . By a similar discussion, we can show that $L (A \to B) \supseteq L (A^{'} \to B^{'})$ . Hence $L (A \to B) = L (A^{'} \to B^{'})$ and so (L, → , E) forms a distributive extended implicative groupoid. The rest of the proof is similar to the previous part.□

The next example provides an extended implicative groupoid which fulfills the conditions of Theorem 5.6.

Example 5.9 Define the operation → on L = {a, b, c, d} as follows:

→	a	b	c	d
a	d	d	d	d
b	a	d	d	d
c	a	b	c	d
d	a	b	b	d

Then by considering E = {c, d}, it follows that the triple (L, → , E) is an extended implicative groupoid. It is easy to see that (L, ≤) is a complete lattice such that (E, ⪯) attains its greatest lower bound ⊥_E = {c}.

Since for any A, A′, B, B′ ⊆ L, $U (A) = U (A^{'})$ if and only if A = A′ and $L (B) = L (B^{'})$ if and only if B = B′. Then $L (A \to B) = L (A^{'} \to B^{'})$ . Hence, (L, → , E) is distributive. For any x, y ∈ L, we define x ⊗ y = ∧ {z ∈ L : y ≤ x → z} . Then the following diagram describes the adjoint product.

⊗	a	b	c	d
a	a	a	a	a
b	a	b	b	b
c	a	b	c	d
d	a	b	d	d

Hence (L, ⊗) is commutative. On the other hand, since for any x, y, z ∈ L, by Proposition 5.1(vi), x ⊗ (y ⊗ z) = (x ⊗ y) ⊗ z, we have (L, → , E) is associative.

Define the operation ⇝ on L = {a, b, c, d} similar to the operation →. Then the implications → and ⇝ form a Galois pair [→ , ⇝], since for any x, y, z ∈ L, x ≤ y → z if and only if y ≤ x ⇝ z. Hence (L, → , E) is a symmetrical extended implicative groupoid.

Consequently, since (L, → , E) is a symmetrical associative complete distributive extended implicative groupoid such that (E, ⪯) attains its greatest lower bound ⊥_E = {c}, by Theorem 5.6, (L, ∧ , ∨ , ⊗ , → , ⇝ , ⊥ _E) is a complete residuated lattice.

6 Conclusion

Let (L, → , E) be a symmetrical associative complete distributive extended implicative groupoid with the adjoint product ⊗ and dual (L, ⇝ , E). We proved that (L, ⊗) is a monoid with unit ⊥_E ∈ E and the residuation properties are satisfied if (E, ⪯) attains its greatest lower bound ⊥_E ∈ E. Consequently (L, ≤ , ⊗ , ⊥ _E) satisfies all of the properties of complete residuated lattices. Thus extended implicative groupoids are useful constructions which strictly include extended order algebras while they can satisfy almost all of their good properties (such as the existence of MacNeille completion), under additional assumptions that the underlying upper bound set E contains both its lower and upper bounds. Hence no (much) loss of generality is available in replacing the existed singleton {⊤} in the settings of extended order algebras with some non-empty upper bound set E, especially when E attains both its lower and upper bounds.

References

Aaly Kologani

, Jun

Y.B.

, Xin

X.L.

, Roh

E.H.

and Borzooei

R.A.

, On co-annihilators in hoops, Journal of Intelligent and Fuzzy Systems 37 (2019), 5471–5485.

Blount

and Tsinakis

, The structure of residuated lattices, International Journal of Algebra and Computation 13(4) (2003), 437–461.

Borzooei

R.A.

, Hosseini

and Zahiri

, Convex L-lattice subgroups in L-ordered groups, Categories and General Algebraic Structures with Applications 9(1) (2018), 139–161.

Borzooei

R.A.

, Hosseini

and Zahiri

, (Totally) L-ordered groups, Journal of Intelligent and Fuzzy Systems 30 (2016), 1489–1498.

Borzooei

R.A.

, Mobini

and Ebrahimi

M.M.

, The category of soft sets, Journal of Intelligent and Fuzzy Systems 28 (2015), 157–177.

Borumand Saeid

, Flaut

, Hoskova-Mayerova

, Afshar

and Kuchaki Rafsanjani

, Some connections between BCK-algebras and n-ary block codes, Soft Computing 22 (2018), 41–46.

Cignoli

R.L.

, D’Ottaviano

I.M.

and Mundici

, Algebraic foundations of many-valued reasoning, Kluwer, Dordrecht, (2000).

Della Stella

M.E.

and Guido

, Associativity, commutativity and symmetry in residuated structures, Order 30(2) (2013), 363–401.

Flaut

, Hoskova-Mayerova

, Borumand Saeid

A.B.

and Vasile

, Wajsberg algebras of order n (n ≤ 9), Neural Computing and Applications 32 (2020), 1330–13312.

10.

Galatos

, Jipsen

, Kowalski

and Ono

, Residuated lattices: An algebraic glimpse at substructural logics, Studies in Logic and the Foundations of Mathematics, 151, Elsevier, Amsterdam, (2007).

11.

Guido

, Relational groupoids and residuated lattices, TACL 2013, EPiC Series, 25, 92–95.

12.

Guido

and Toto

, Extended-order algebras and fuzzy implicators, Soft Computing 16 (2012), 1883–1892.

13.

Guido

and Toto

, Extended-order algebras, Journal of Applied Logic 6 (2008), 609–626.

14.

Hájek

, Metamathematics of fuzzy logic, Trends in Logicstudia Logica Library, 4, Kluwer, Dordrecht, (1998).

15.

Jipsen

and Tsinakis

, A survey of residuated lattices, Ordered Algebraic Structures, Kluwer, Dordrecht, (2002), 19–56.

16.

Rasiowa

, An algebraic approach to non-classical logics, Studies in Logics and the Foundations of Mathematics, 78, North-Holland, Amsterdam, (1974).