Abstract
In the present paper, we introduce the notion of q′-compact in Łukasiewicz near semirings and we prove that if induced lattice with an antitone involution of a Łukasiewicz near semiring is a complete lattice and q′-compact, then the induced lattice is a strongly algebraically closed lattice. Also, it is shown that every Łukasiewicz near semiring is representable as a Sheffer stroke basic algebra. In the final part of the article, we prove that the variety of Sheffer stroke basic algebras is congruence regular and arithmetical.
Keywords
Introduction
More recently, the relations between prominent algebraic structures from many-valued logics and (semi)ring theory have stirred a renewed attention (see [1, 13]). In particular, it was shown by Belluce, Di Nola, Ferraioli [1] and Gerla [13] that MV-algebras can be represented as semirings but basic algebras can not be represented as semirings. These observations seem to suggest that a substantial weakening of the concept of semiring would be required to embrace such algebras. In order to provide a semiring-like representation of basic algebras, S. Bonzio, I. Chajda and A. Ledda in [2] specialized the concept of near semiring and introduced the notion of Łukasiewicz near semiring and orthomodular near semiring and proved that basic algebras and orthomodular lattices can be represented as Łukasiewicz near semirings. In this work, we show that every Łukasiewicz near semirings is representable as a Sheffer stroke basic algebra.
Recall that the Sheffer stroke term operation was firstly given by H. M. Sheffer in [17] and Sheffer stroke basic algebra is defined in [18]. In the following, we introduce the notion of q′-compactness in Łukasiewicz near semirings and we obtain the conditions such that induced lattice with an antitone involution of a Łukasiewicz near semiring is a strongly algebraically closed lattice.
Recall, in [21] J. Schmid proved that a distributive lattice is a algebraically closed lattice if and only if it is a Boolean lattice. Also, he shows that any strongly algebraically closed lattice is a complete Boolean lattice. Later, it is proved by the author in [16] that if a complete Boolean lattice is q′-compact, then it is a strongly algebraically closed lattice. We recall from [16] that an algebra A is strongly algebraically closed in a class of algebras, if every set of equations (finite or infinite) with coefficients from A, which is solvable in some algebras of the class of algebras containing A, already has a solution in A.
The paper is organized as follows. In Section 2, we review some definitions and results which will be used in the following. In Section 3, it is shown that every Łukasiewicz near semiring can be represented by a Sheffer stroke basic algebra. In Section 4, we introduce the notion of q′-compactness for Łukasiewicz near semirings and prove that if induced lattice with an antitone involution of a Łukasiewicz near semiring is a complete lattice and q′-compact, then the induced lattice is a strongly algebraically closed lattice. Finally, in section 5, it is proved that the variety of Sheffer stroke basic algebras is congruence regular and arithmetical.
Preliminaries
We recall of [4] that a join-semilattice (meet-semilattice) is a groupoid satisfying identities: idempotency, associativity, commutativity. Join-semilattices (Meet-semilattices) are in one-to-one correspondence with posets any two elements of which have a supremum (infimum).
Suppose that (A ; ∨ , ∧) is an algebra of type (2, 2) and (A ; ∨) is a joinsemilattice and (A ; ∧) is a meet-semilattice. If (A ; ∨ , ∧) satisfies the identities (x ∨ y) ∧ x = x and (x ∧ y) ∨ x = x, then (A ; ∨ , ∧) is called a lattice, for all x, y, z ∈ A.
A lattice (L ; ∨ , ∧) is distributive when it satisfies for all x, y, z ∈ L:
Let (A ; ∨ , ∧) be an algebra. A mapping g from A to A is said isotone (or monotone) if x, y ∈ A and
Also, the mapping g is antitone if x, y ∈ A and
Assume that for each a in a poset (A ; ≤) with greatest element 1 there exists a mapping (or antitone involution) g a : [a, 1] ⟶ [a, 1] with the following property:
(1) g a (g a (x)) = x, for all x ∈ [a, 1];
(2) x ≤ y ⇒ g a (y) ≤ g a (x).
We will write x a instead of g a (x) and for a ∈ A the intervals [a, 1] will be called sections of (A ; ≤) and the mappings x ⟶ x a for every x ∈ [a, 1] will be said section antitone involutions. Suppose that (L ; ∨ , ∧) a lattice with a greatest element 1. The structure (L ; ∨ , ∧ , ( a ) a∈A, 1) will be called a lattice with section antitone involutions, if its induced poset (L ; ≤) is a poset with section antitone involutions.
A bounded lattice is a lattice (L ; ∨ , ∧ , 1) with a least element 0 and a greatest element 1. Lattice L is equipped with complementation if there exists a mapping ⟶x′ of L into L such that
If there is a mapping x ⟶ x′ of L into itself such that x′′ = x and x ≤ y ⇒ y′ ≤ x′, then we say that L is a lattice with an antitone involution. Recall that a lattice with an antitone involution L satisfies the so-called de Morgan Laws for all a, b ∈ L:
By a Boolean lattice we mean a complemented distributive lattice. By a Boolean algebra we mean a Boolean lattice together with the unary operation of complementation.
Łukasiewicz near semirings and Sheffer stroke basic algebras
In this section, we discuss the relations between Łukasiewicz near semirings and Sheffer stroke basic algebras. For the convenience of the reader we recall some useful concepts.
(i) (R ; + , 0) is a commutative monoid;
(ii) (R ; · , 1) is a groupoid satisfying x · 1 = x = 1 · x (a unital groupoid);
(iii) (x + y) · z = (x · z) + (y · z);
(iv) x · 0 =0 · x = 0;
for all x, y, z ∈ R.
We recall from [2] that a semiring is a near semiring such that (R ; · , 1) is a monoid (i.e. · is also associative) that satisfies left distributivity: x · (y + z) = (x · y) + (x · z), for all x, y, z ∈ R.
A near semiring is called idempotent if it satisfies x + x = x. It is clear that (R ; +) is a semilattice. In particular, (R ; +) can be considered as a join-semilattice, where the induced order is defined as x ≤ y iff x + y = y and the constant 0 is the least element ([2], Remark 1).
(1) α (α (x)) = x;
(2) if x ≤ y then α (y) ≤ α (x).
As we mentioned in the introduction this article, S. Bonzio, I. Chajda and A. Ledda in [2], in order to provide a semiring-like representation of basic algebras, specialized the concept of near semiring and introduced the notion of Łukasiewicz near semiring.
α (x · α (y)) · α (y) = α (y · α (x)) · α (x).
As we mentioned, we study the relations between Łukasiewicz near semirings and Sheffer stroke basic algebras and we will show that every Łukasiewicz near semiring can be represented by a Sheffer stroke basic algebra.
(1) x|y = y|x;
(2) (x|x) | (x|y) = x ;
(3) x| ((y|z) | (y|z)) = ((x|y) | (x|y)) |z;
(4) (x| ((x|x) | (y|y))) | (x| ((x|x) | (y|y))) = x;
for all x, y, z ∈ A .
(SH1) (x| (x|x)) | (x|x) = x;
(SH2) (x| (y|y)) | (y|y) = (y| (x|x)) | (x|x);
(SH3) (((x| (y|y)) | (y|y)) | (z|z)) | ((x|) z|z)) | (x| (z|z))) = x| (x|x); for all x, y, z ∈ A .
Basic algebras are an important concept used in different non-classical logics and were introduced in [5–10]. Basic algebras introduced in the last decade by Hala ¡s, K¨uhr, and Chajda, as a common generalization of both MV-algebras and orthomodular lattices. They can be regarded as a non-associative and non-commutative generalization of MV algebras. These algebras are in bijective correspondence with bounded lattices having an antitone involution on every principal filter (sectional antitone involutions).
(1) x ⊕ 0 = x;
(2) (x′ )′= x;
(3) (x′ ⊕ y) ′ ⊕ y = (y′ ⊕ x) ′ ⊕ x;
(4) (((x⊕ y) ′ ⊕) ′ ⊕ z) ′ ⊕ (x ⊕ z) =1 ;
for all x, y, z ∈ A .
On every basic algebra, a partial order ≤ on A is defined by the rule
Now, we are ready to prove the following theorem.
x|y = α ((α (α (x)) + α (y)) · α (α (y))), for all x, y ∈ R .
Using ([18], Definition 3.1), we prove that (R ; |) satisfies in the definition of Sheffer Stroke basic algebra. We have:
Therefore, condition (SH1) of definition of Sheffer stroke basic algebra satisfies. Now, about condition (SH2) of definition of Sheffer stroke basic algebra we have:
Thus,
Similarly, condition (SH3) of definition of Sheffer stroke basic algebra satisfies. □
We know that α (x) = x′ = x|1 and thus we will have that:
Similarly, about operation “·" have that:
Using ([2], Theorem 5), completes the proof. □
Strongly algebraically closed Łukasiewicz near semirings
System of equations we mean an arbitrary set of equations. Recall from [20] that Boolean algebras B1, B2 are geometrically equivalent if for any system of equations the coordinate algebras over B1 and B2 are isomorphic. This means that a description of coordinate algebras over an algebra B1 automatically implies the corresponding description over any algebra B2 which is geometrically equivalence to B1 [12]. We recall that the problem “When two geometrically equivalent extensions L1 and L2 of a field P have different elementary theories in the logik?" of the study of geometric equivalence was posed in [19]. In [15] this problem was solved for equationally Noetherian groups. Theorem 7.2 of [20] contains a criterion for a pair of boolean algebras to be geometrically equivalent. Following this, in the present paper we define geometrically equivalence for Łukasiewicz near semirings, and we try to establish a relationship between Łukasiewicz near semirings and the strongly algebraically closed algebras.
A lattice A in a class is said to be strongly algebraically closed if every system (not necessarily finite) of equations with parameters in A which has a solution in some extension of A, has already a solution in A.
Equationally and weak equationally Noetherian boolean algebras (with coefficients) are characterized by Shevlyakov in [20]. As it is shown in [20], a boolean algebra A is equationally Noetherian if and only if it is finite; and it is weak equationally Noetherian if and only if it is complete. A lattice A is called equationally Noetherian, if any system of equations with coefficient in A is equivalent with a finite subsystem. If any system of equations over A is equivalent with a finite system then it is said weakly equationally Noetherian.
Now, we ready to prove the following theorem.
On every basic algebra, a partial order ≤ on A is defined by the rule
We shall use the notation:
It remains that to prove that (A ; ∨ , ∧ 0, 0, 1) is a strongly algebraically closed lattice. Let S be a consistent system in the language
On the other hand, we have
Since T is finite, we also have
This shows that S and T are equivalent over B. Therefore, T has a solution in B and consequently in A. Thus S has a solution in A. Note that T is a finite system in the language
1- If T contains the Boolean constants 0 and 1, then there will be no change, since 0, 1 ∈ A.
2- If T contains therm x′, then we introduce a new variable y and insert new equations x ∧ y ≈ 0 and x ∨ y ≈ 1, instead.
3- If there appears a term of the form a′, then again there will not be any changes. □
Congruence properties of Sheffer stroke basic algebras
Regular varieties have been characterized independently by B. Csakany, G. Gratzer and R. Wille in 1970s. For our purposes we present a Mal’cev condition which is rather similar to that one of R. Wille (cf. Theorem 6.11 in [22]).
By a congruence of a partial structure (A ; F) we mean an equivalence relation r on A such that for any n-ary operation symbol F, (a1, b2) ∈ r, …, (a n , b n ) ∈ r imply (F A (a1, …, a n ), F A (b1, …, b n )) ∈ r whenever F A (a1, …, a n ) and F A (b1, …, b n ) are both defined (substitution property).
An algebra A is said to be congruence regular if any two congruences of A with a common block are equal. A variety is said to be congruence regular if all its algebras are.
An algebra (A ; F) is said congruence distributive if the lattice Con (A) is distributive. A variety V of algebras is said congruence distributive if every member of V is congruence distributive.
An algebra (A ; F) is said to have permutable congruences if ros = sor for any pair r, s of congruences of A. A variety V is said to have permutable congruences (or to be congruence permutable) if every algebra in V has permutable congruences.
We recall form [14] that a variety V is congruence regular if and only if there exists a set of ternary terms t i (x, y, z) with i ≥ 1 such that t i (x, y, z)=z for any i if and only if x = y.
Finally, an algebra A is arithmetical if it is both congruence permutable and congruence distributive. A variety V is arithmetical if each algebra A ∈ V is arithmetical.
Now we prove that t2 (x, x, z) = z.
t2 (x, x, z) = (((((x| (x|1)) |1) | (((x| (x|1)) |1) |1)) | (((x|(x|1)) |1) |1) |1) |z) |1 = (((((0) |1) | ((0)) |1)) | ((0) |1)) |z) |1 = 1 . z = (1|z) |1 = z,
and then t2 (x, x, z) = z.
Now suppose that t1 (x, y, z) = t2 (x, x, z), then we prove that x = y. We set
From (*) we have that a ≤ z, and z|z ≤ a|a. Let us check that that a = 0 .
Similarly,
Therefore, this variety is congruence permutable. Moreover, the following ternary term
This proves that the variety of Sheffer stroke basic algebras is arithmetical.□
1- Every basic algebra can be represented by Łukasiewicz near semirings and so it is represented by a Sheffer stroke basic algebra.
2- Suppose x ⟶
K
y is n + 1-valued function defined in the following way:
Conclusion
Łukasiewicz semirings are objects of prominent importance for algebraic logic. In fact, MV-algebras, the equivalent algebraic semantics of Łukasiewicz many-valued logic, can be represented in terms of Łukasiewicz semiring and here we show that every Łukasiewicz semiring can be represented a stroke basic algebra. In this work we introduce the notion of q′-compact in Łukasiewicz near semirings and we made the first steps toward the description of Łukasiewicz logic, establish a link with strongly algebraically closed lattices. Anyway we also note that new fields like many-valued formal languages could find in this framework the more appropriate ground where to develop.
Footnotes
Acknowledgments
We thank the anonymous referee for his/her valuable comments and hints which made our paper more readable.