Fuzzy admissible congruences on type B semigroups with E -properties,and their applications

Abstract

Type B semigroups are described as the generalized inverse semigroups in the range of abundant semigroup. Motivated by studying fuzzy congruences in inverse semigroups, and as a continuation of N. Kuroki’s work in inverse semigroups and our work in abundant semigroups in terms of fuzzy subsets, this paper considers fuzzy admissible congruences on some classes of type B semigroups. Our main purpose is to show when a fuzzy admissible congruence on a type B semigroup with E-properties is E-properties preserving. In particular, we get some sufficient and necessary conditions for some classes of type B semigroups to be primitive, E-unitary and E-reflexive, respectively. As an application, we extend our results to the cases of inverse semigroups.

Keywords

Fuzzy admissible congruences type B semigroups primitive

AMS Mathematics Subject Classification (2010) 06F05; 20M10

1 Introduction

As a generalization of classical sets, a fuzzy set was presented by Zadeh [34] to characterize uncertainty in 1965. Following [34], a fuzzy subset f of a non-empty set X is a function of X into the closed interval [0, 1]. Since then, a series of research on fuzzy sets has come out expounding the importance of the concept and its applications to logic, algebra theory, real analysis, topology, etc ([2 , 35]). In 1971, Rosenfeld [29] introduced the notion of fuzzy subgroup. Following the formulation of fuzzy groups [29], many authors have been engaged in extending the notions of pure algebra to the broader framework of the fuzzy set ([7 , 32]). In semigroups, after Kuroki’s work [14], Murali [25], Nemitz [26], Samhan [30] and others [15, 18] define fuzzy equivalent relations and fuzzy congruences on a semigroup, respectively. By Nemitz [26], for a non-empty set X, a fuzzy relation μ on X is a map from X × X into the interval [0, 1]. Let S be a semigroup and μ, ν be two fuzzy relations on S. Then the product μ ∘ ν of μ and ν is defined by the rule μ ∘ ν (a, b) = ∨ _x∈S [μ (a, x) ∧ ν (x, b)] for all a, b ∈ S, and μ ⊆ ν is defined by μ (x, y) ≤ ν (x, y) for all x, y ∈ S . As in [24], a fuzzy relation μ on S is fuzzy equivalent if μ (a, a) =1, μ (a, b) = μ (b, a) and μ ∘ μ ⊆ μ hold for all a, b ∈ S. A fuzzy relation μ on S is compatible if μ (ax, bx) ≥ μ (a, b) and μ (xa, xb) ≥ μ (a, b) hold for all a, b, x ∈ S . A fuzzy equivalence relation μ which is compatible on S is called a fuzzy congruence on S . Let μ be a fuzzy equivalence relation on S . For each a ∈ S, a fuzzy subset μ_a of S is defined by the rule μ_a (x) = μ (a, x) for all x ∈ S . In particular, if μ is a fuzzy congruence on S, then S/μ = {μ_a | a ∈ S} is a semigroup with respective to the binary operation “★” on S/μ by the rule μ_a ★ μ_b = μ_ab. In 1997, kuroki [15] studied fuzzy congruences on inverse semigroups (a semigroup S is inverse if S is regular and the idempotents of S commute). Recently, many authors are engaged in studying various classes of fuzzy semigroups, and get many interesting results ([1 , 36]).

Following [6], for a semigroup S, two elements a and b of S are related by lx [resp., rx] on S if and only if they are related by lc [resp., rc] in a semigroup T containing S (see, [6]). In other words, the relations lx and rx are generalized Green’s relations lc and rc, respectively. The intersection of the equivalence relations lx and rx is denoted by hx (i.e., hx = lx ∩ rx). Recall that a semigroup S is said to be right abundant [resp., left abundant] if each lx class [resp., rx class] of S contains an idempotent. A right [resp., left] abundant semigroup in which the idempotents commute is right adequate [resp., left adequate]. A semigroup S is abundant [resp., adequate ] if it is both right and left abundant [resp., both right and left adequate]. It can be easily seen that each lx class [resp., rx class] of an adequate semigroup S contains exactly one idempotent. As usual, a⁺ [a^*] denotes a typical idempotent rx-related [lx-related] to a ; E (S) denotes the set of idempotents of S. In [16], Lawson defined a natural partial order relation “≤” on an abundant semigroup S by the rule a ≤ b if and only if there are e, f ∈ E (S) , such that a = eb = bf for all a, b ∈ S . As in [6], a right adequate semigroup S is right type B, if it satisfies Conditions (B1) and (B2):

(∀ e, f ∈ E (S¹) , a ∈ S) (efa) ^* = (ea) ^* (fa) ^* ;

(∀ a ∈ S, e ∈ E (S)) e ≤ a^* ⇒ (∃ f ∈ E (S¹)) e = (fa) ^* .

A left type B semigroup is defined in the dual way. A semigroup S is type B if it is both right and left type B. Fountain, El-Qallali, Gould and others study various classes of abundant semigroups (see, [4–6 , 19–21]).

Motivated by N. Kuroki’s work in an inverse semigroup in terms of fuzzy subsets, and as a continuation of our work in fuzzy abundant subsemigroups, we attempt in this paper to study the fuzzy congruences on some classes of type B semigroups in detail. As we know, an arbitrary admissible congruence ρ (a congruence ρ on a semigroup S is admissible if and only if aρ lx (S/ρ) a^*ρ and aρ rx (S/ρ) a⁺ρ for all a ∈ S) on an abundant semigroup is abundant lx-class preserving and abundant rx-class preserving. In [18], we introduced the notion of a fuzzy admissible congruence (a fuzzy congruence μ on an abundant semigroup S is admissible if μ_a lx (S/μ) μ_{a
^*} and μ_a rx (S/μ) μ_{a
⁺} for all a ∈ S) on an abundant semigroup which preserves the relation lx and rx with respect to the binary operation “ ★”. In this paper, we shall consider fuzzy admissible congruences on some classes of type B semigroups by using the results of [18]. Our main goal is to show the relations between fuzzy congruences and semigroup algebraic structure. It is proved that there exist many important relations between fuzzy admissible congruences on a type B semigroup and its E-properties such as E-unitary, E-reflexive, and so on. As an application, our results can be used to describe inverse semigroups with E-properties.

2 Preliminaries

We follow the notions adopted in [4 , 27]. First, we state some known results which will be frequently used throughout the paper.

Lemma 2.1. [4] Let S be a semigroup and a, b ∈ S. Then the following statements are equivalent:

alxb [arxb];

for all x, y ∈ S¹, ax = ay [xa = ya] if and only if bx = by [xb = yb].

Lemma 2.2. [4] Let S be a semigroup and a ∈ S, e ∈ E (S). Then alxe [arxe] if and only if ae = a [ea = a] and for all x, y ∈ S¹, ax = ay [xa = ya] implies ex = ey [xe = ye].

It is easy to check that lc ⊆ lx and rc ⊆ rx, and that lx [resp., rx] is a right [resp., left] congruence. For any regular elements a and b, (a, b) ∈ lx [resp., (a, b) ∈ rx] if and only if (a, b) ∈ lc [resp., (a, b) ∈ rc]. In other words, for an arbitrary regular semigroup S, lx = lc, rx = rc and hx = hc . Recall from [27] that a congruence ρ on a semigroup S is idempotent-separating, if for all e, f ∈ E (S) the equality eρ = fρ implies e = f . A congruence ρ on a semigroup S is cancellative if S/ρ is a cancellative semigroup. Evidently, a cancellative congruence is admissible. From [4], we quote

Lemma 2.3. [4] Let S be an abundant semigroup. Then the following statements are true:

η_l = {(a, b) ∈ S × S | (∀ e ∈ E (S)) ealxeb} is the largest congruence on S contained in lx . The dual result holds for η_r ;

η = η_l ∩ η_r is the largest congruence on S contained in hx .

For a type B semigroup S, η is the largest idempotent-separating admissible congruence on S contained in hx . In [17], we defined the least cancellative monoid congruence σ on a type B semigroup S by the rule $(\forall a, b \in S) a σ b \Leftrightarrow (\exists e \in E (S)) (eae = ebe) .$

Next, we recall some of the basic facts about the fuzzy relations.

Let S be a semigroup. A fuzzy congruence μ on S is idempotent-separating [resp., idempotent-pure] if for all e, f ∈ E (S) , a ∈ S the equality μ_e = μ_f [resp., μ_a = μ_e] implies e = f [resp., a ∈ E (S)]. Denote by C_ρ the characteristic function of a binary relation ρ on S . Then, as is easily seen, ρ is a congruence on S if and only if C_ρ is a fuzzy congruence on S .

Lemma 2.4. [24] Let μ be a fuzzy equivalence relation on a semigroup S. Then μ_a = μ_b ⇔ μ (a, b) =1 for all a, b ∈ S .

Obviously, if μ is a fuzzy admissible congruence on an abundant semigroup S, then S/μ is an abundant semigroup for operation “★.” In particular, if S/μ is a cancellative semigroup for operation “★”, then we call μ a fuzzy cancellative congruence. For brevity, we denote by (μ_a) ⁺ [resp., (μ_a) ^*] a typical idempotent rx-related [resp., lx-related] to μ_a .

The following lemma is due to our work in [18].

Lemma 2.5. [18] Let ρ be a congruence on an abundant semigroup S. Then the following statements are true:

ρ is admissible on S if and only if C_ρ is a fuzzy admissible congruence on S;

if μ is a fuzzy admissible congruence on S such that μ ⊆ C_hx, then μ is an idempotent-separating fuzzy admissible congruence on S;

if S is adequate, then (i) S/μ is adequate for operation “★”; (ii) μ_a ∈ E (S/μ) ⇔ (∃ e ∈ E (S)) μ_a = μ_e; (iii) (∀ a ∈ S) μ_a
⁺ = (μ_a) ⁺ and μ_a
^* = (μ_a) ^* ;

if S is adequate, then μ is fuzzy cancellative if and only if μ_e = μ_f for all e, f ∈ E (S) .

3 Some basic properties of fuzzy admissible congruences on type B semigroups

In this section, we shall consider fuzzy admissible congruences on type B semigroups. It is proved that an arbitrary fuzzy admissible congruence on type B semigroup admits the “ type B” property.

Lemma 3.1. Let μ be a fuzzy admissible congruence on an adequate semigroup S and a, b ∈ S . Then the following statements are true:

a≤ b ⇒ μ_a ≤ μ_b ;

μ_a ≤ μ_b ⇒ μ_a
⁺ ≤ μ_b
⁺, μ_a
^* ≤ μ_b
^* .

Proof. (1) It follows immediately from Lemma 2.5(3) and the definition of “≤”.

(2) By Lemma 2.5(3), E (S/μ) = {μ_e | e ∈ E (S)} . Let μ_a, μ_b ∈ S/μ such that μ_a ≤ μ_b . Then μ_a = μ_e ★ μ_b = μ_b ★ μ_f, where e, f ∈ E (S) . That is, μ_a = μ_eb = μ_bf . Hence, by Lemma 2.5(3), μ_{a
⁺} = (μ_a) ⁺ = (μ_eb) ⁺ = μ_(eb)⁺ = μ_{eb
⁺} = (μ_bf) ⁺ = μ_(bf)⁺ = μ_b⁺(bf)⁺ since S is rx-unipotent (i.e., each rx-class contains exactly one idempotent). Thus, μ_{a
⁺} = μ_e ★ μ_{b
⁺} = μ_{b
⁺} ★ μ_(bf)⁺ . That is, μ_{a
⁺} ≤ μ_{b
⁺}. Similarly, we can prove that μ_{a
^*} ≤ μ_{b
^*}. This completes the proof.□

Theorem 3.2. Let μ be a fuzzy admissible congruence on a type B semigroup S. Then S/μ is a type B semigroup for operation “★.”

Proof. Let μ be a fuzzy admissible congruence on S. Then, by Lemma 2.5(3), S/μ is an adequate semigroup for operation “★.” Next, we prove that S/μ is type B. To see it, let μ_e, μ_f ∈ E (S/μ) and μ_a ∈ S/μ, where e, f ∈ E (S) . Then $\begin{matrix} (μ_{e} ★ μ_{f} ★ μ_{a})^{*} & = & (μ_{efa})^{*} = μ_{(efa)^{*}} (by Lemma 2.5 (3)) \\ = & μ_{(ea)^{*} (fa)^{*}} (since S satisfies Condition (B 1)) \\ = & μ_{(ea)^{*}} ★ μ_{(fa)^{*}} \\ = & (μ_{ea})^{*} ★ (μ_{fa})^{*} (by Lemma 2.5 (3)) \\ = & (μ_{e} ★ μ_{a})^{*} ★ (μ_{f} ★ μ_{a})^{*} . \end{matrix}$ This gives that S/μ satisfies Condition (B1). On the other hand, let μ_a ∈ S/μ, μ_e ∈ E (S/μ) such that μ_e ≤ (μ_a) ^*, where e ∈ E (S) . Then, by Lemma 2.5(3), μ_e ≤ μ_{a
^*} . That is, μ_e = μ_e ★ μ_{a
^*} = μ_{ea
^*} . But ea^* ≤ a^* . We get that ea^* = (fa) ^* for some f ∈ E (S) since S satisfies Condition (B2). Hence, μ_e = μ_{ea
^*} = μ_(fa)^* = (μ_fa) ^* = (μ_f ★ μ_a) ^*, where μ_f ∈ E (S/μ) , and thus S/μ satisfies Condition (B2). Therefore, S/μ is right type B. Dually, we can prove that S/μ is left type B. This means that S/μ is a type B semigroup for operation “★.” □

We now give three examples of a type B semigroup which satisfies the conditions in Theorem 3.2.

Example 3.1. Let N be the set of all non-negative integers and S = {(m, n) ∈ N × N | m ≥ n} . Define a multiplication “•” on S by $(m, n) • (p, q) = (m - n + t, q - p + t), where t = \max {n, p} .$ Then, it is easy to check that (S, •) is a semigroup with E (S) = {(m, m) ∈ N × N} , and that for all (m, n) ∈ S, there exist idempotents (n, n) , (m, m) of S such that $(m, n) L (n, n) and (m, n) R (m, m) .$ In fact, it is easy to see that S is type B. Hence, (m, n) ^* = (n, n) and (m, n) ⁺ = (m, m) .

Consider a relation δ on S as follows: $(m, n) δ (p, q) \Leftrightarrow m - n = p - q for (m, n), (p, q) \in S .$ Obviously, δ is a congruence. Moreover, it is easy to see that (m, n) δlx (n, n) δ and (m, n) δrx (m, m) δ . Therefore, δ is an admissible congruence on S .

By Lemma 2.5(1), C_δ is a fuzzy admissible congruence on S . Now, we prove that S/C_δ is type B for operation “★.” Note that S/C_δ is an abundant semigroup for operation “★.” Moreover, we can see easily that S/C_δ is adequate for operation “★”, and that E (S/C_δ) = {(C_δ) _(m, m) | m ∈ N} . On the other hand, it is clear that (C_δ) _(m, m) = (C_δ) _(n, n) for all (m, m) , (n, n) ∈ E (S) . Therefore, we can see easily that S/C_δ satisfies Conditions (B1) and (B2). That is, S/C_δ is right type B. Similarly, we can prove that S/C_δ is left type B.

Summarizing up the above arguments, we conclude that S/C_δ is type B.

Example 3.2. Let N be the set of all non-negative integers and S = { 2ⁿ | n ∈ N}. Then it is easy to see that S is a non-regular type B monoid for the general multiplication such that a^* = a⁺ = 1 for all a ∈ S . Define a fuzzy relation on S as follows: $μ : S \times S ⟶ [0, 1],$

$μ (a, b) = {\begin{matrix} 1 & a = b \\ (1 - 1 / a) \land (1 - 1 / b) & a \neq b, a \neq 1 \neq b \\ 1 - 1 / b & a = 1, b \neq 1 \\ 1 - 1 / a & b = 1, a \neq 1 \end{matrix}$

It is easily seen that μ is a fuzzy congruence on S, and that (μ_a) ^* = (μ_a) ⁺ = μ₁ for all a ∈ S . Next we prove that μ is fuzzy admissible. To see it, let μ_a, μ_x, μ_y ∈ S/μ be such that μ_a ★ μ_x = μ_a ★ μ_y, that is, μ_ax = μ_ay. Then μ_ax (1) = μ_ay (1). Hence, μ (ax, 1) = μ (ay, 1). Consider the following two possible cases:

if a = 1, then μ_x = μ_ax = μ_ay = μ_y . That is, μ₁★ μ_x = μ₁ ★ μ_y ;

if a ≠ 1, then ax ≠ 1 ≠ ay . Hence, we have that μ (ax, 1) = μ (ay, 1) implies 1 - 1/ax = 1 -1/ay . Thus x = y, and so μ₁ ★ μ_x = μ₁ ★ μ_y .

Summarizing up the above Cases (1) and (2). We have μ_a lx (S/μ) μ₁ . Similarly, μ_a rx (S/μ) μ₁ . Therefore, μ is fuzzy admissible.

On the other hand, we can see easily that S/μ satisfies Conditions (B1) and (B2). That is, S/μ is right type B. Similarly, we can prove that S/μ is left type B. This gives that S/μ is type B for operation “★.”

Example 3.3. Let $S = {[x]_{2 \times 2} | x \in ℕ} \cup {[\frac{1}{2}]_{2 \times 2}}$ , where N is the set of all non-negative integers and [x] _2×2 denotes the following second-order matrix

$(\begin{matrix} x & x \\ x & x \end{matrix})$ for all $x \in ℕ \cup {\frac{1}{2}}$ . Then it is easy to see that S is an abundant semigroup with respect to the general matrix multiplication, and that $E (S) = {(\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}), (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{matrix})} .$ In fact, it is easily seen that lx-classes and rx-classes of S are both S \ {[0] _2×2} and {[0] _2×2}. Obviously, E (S) is a semilattice and $[a]_{2 \times 2}^{*} = [a]_{2 \times 2}^{+} = [\frac{1}{2}]_{2 \times 2}$ for all a ∈ N and a ≠ 0. Thus, it is routine to check that S satisfies the Conditions (B1) to (B2) and dual conditions of Conditions (B1) and (B2). Therefore, S is a type B semigroup. Define a fuzzy relation on S as follows: $μ : S \times S ⟶ [0, 1],$ $μ ([a]_{2 \times 2}, [b]_{2 \times 2}) = {\begin{matrix} 1 & a = b \\ 0 & a \neq b, ab = 0 \\ (1 - 1 / a) \land (1 - 1 / b) & a \neq b, ab \neq 0, a \neq \frac{1}{2} \neq b \\ \frac{1}{2} \land (1 - 1 / b) & a = \frac{1}{2}, ab \neq 0, b \neq \frac{1}{2} \\ (1 - 1 / a) \land \frac{1}{2} & a \neq \frac{1}{2}, ab \neq 0, b = ab \neq 0 \end{matrix}$

It is easy to see that μ is a fuzzy congruence on S . Furthermore, we have (μ_{[0]_2×2}) ^* = (μ_{[0]_2×2}) ⁺ = μ_{[0]_2×2} and $(μ_{[a]_{2 \times 2}})^{*} = (μ_{[a]_{2 \times 2}})^{+} = μ_{[\frac{1}{2}]_{2 \times 2}}$ for all a ≠ 0 . By a similar method of Example 3.2, we have that μ is fuzzy admissible, and that S/μ is type B for operation “★.”

4 Fuzzy admissible congruences on type B semigroups with E-properties

In this section, we shall consider fuzzy admissible congruences on some classes of type B semigroups with E-properties. Recall that a semigroup S is left E-unitary if (∀ e ∈ E (S) , a ∈ S) ea ∈ E (S) implies a ∈ E (S) . Dually, we can define right E-unitary. A semigroup S is E-unitary if it is both left and right E-unitary. A type B semigroup S is fundamental [resp., E-reflexive] if η^S = 1_S [resp., (∀ e ∈ E (S) , x, y ∈ S) exy ∈ E (S) ⇒ eyx ∈ E (S)]. The main purpose of this section is to prove when a fuzzy admissible congruence on a type B semigroup with E-properties (e.g., E-unitary, E-reflexive) is E-properties preserving.

Proposition 4.1. Let S be a primitive (i.e., (∀ 0 ≠ e, f ∈ E (S)) e ≤ f ⇒ e = f) type B semigroup. Then S/C_η is primitive for operation “★.”

Proof. By Lemma 2.5 and Theorem 3.2, we have that S/C_η is a type B semigroup for operation “ ★ ″, and so E (S/C_η) = {(C_η) _e | e ∈ E (S)} . Next, we prove that S/C_η is primitive. To see it, let (C_η) _e, (C_η) _f ∈ E (S/C_η) such that (C_η) _e ≤ (C_η) _f, where e, f ∈ E (S) . Then (C_η) _e = (C_η) _e ★ (C_η) _f = (C_η) _f ★ (C_η) _e . That is, (C_η) _e = (C_η) _ef = (C_η) _fe . Note that C_η is an idempotent-separating fuzzy admissible congruence on S. We have that e = ef = fe . Hence, e ≤ f . Thus e = f since S is primitive. Therefore, (C_η) _e = (C_η) _f, as required. □

Remark 4.1. We remark here that the condition “S is a primitive type B semigroup” in Proposition 4.1 can not be weakened to “S is a type B semigroup.” In fact, in our Example 3.1, we have prove that S is a type B semigroup, and it can be easily seen that (2, 2) ≤ (1, 1) for (2, 2) , (1, 1) ∈ E (S) . But (2, 2) ≠ (1, 1) . This gives that S is not primitive. On the other hand, by Lemma 2.5 and Theorem 3.2, S/C_η is type B. Moreover, by Lemma 3.1(1), (C_η) _(2,2) ≤ (C_η) _(1,1) . However, (C_η) _(2,2) ≠ (C_η) _(1,1), for if otherwise, then (C_η) _(2,2) = (C_η) _(1,1) implies that (2, 2) = (1, 1) since C_η is an idempotent-separating fuzzy admissible congruence on S . This contradiction shows that S/C_η is not primitive.

By Theorem 3.2, an arbitrary fuzzy admissible congruence on a type B semigroup has a property of “type B- preserving”. Naturally, one would ask whether every fuzzy admissible congruence on a type B semigroup has a property of “E-properties preserving”? The following theorem will give a positive answer to this problem to the case of an idempotent-pure fuzzy admissible congruence.

Theorem 4.2. Let μ be a fuzzy admissible congruence on a type B semigroup S. Then the following statements are true:

if μ is idempotent-separating, then S/μ is primitive for operation “★” if and only if S is primitive;

if μ is idempotent-pure, then S/μ is E-unitary for operation “★” if and only if S is E-unitary;

if μ is idempotent-pure, then S/μ is E-reflexive for operation “★” if and only if S is E-reflexive;

if μ is cancellative, then S/μ is E-unitary for operation “★”;

S/C_σ is E-unitary for operation “★.”

Proof. (1) Necessity. Let e, f ∈ E (S) with e ≤ f . Then, by Lemma 3.1(1), μ_e ≤ μ_f . But S/μ is primitive. We have μ_e = μ_f . Thus, e = f since μ is idempotent-separating. Therefore, S is primitive.

Sufficiency. By Lemma 2.5(3), E (S/μ) = {μ_e | e ∈ E (S)}. Let μ_e, μ_f ∈ E (S/μ) be such that μ_e ≤ μ_f, where e, f ∈ E (S). Then μ_e = μ_e ★ μ_f = μ_f ★ μ_e . That is, μ_e = μ_ef = μ_fe . Hence, e = ef = fe since μ is idempotent-separating, and so e ≤ f . Note that S is primitive. We have e = f . Thus, μ_e = μ_f, this gives that S/μ is primitive for operation “★.”

(2) Necessity. Let e ∈ E (S) , a ∈ S with ea ∈ E (S) . Then μ_ea ∈ E (S/μ). That is, μ_e ★ μ_a ∈ E (S/μ) . Hence, μ_a ∈ E (S/μ) since S/μ is E-unitary with respect to operation “★.” By Lemma 2.5(3), we have that μ_a = μ_f for some f ∈ E (S) . Hence a ∈ E (S) since μ is idempotent-pure, and so that S is left E-unitary. Similarly, we can prove that S is right E-unitary.

Sufficiency. Note that E (S/μ) = {μ_e | e ∈ E (S)}. Let μ_e ∈ E (S/μ) , μ_a ∈ S/μ, where e ∈ E (S) . If μ_e ★ μ_a ∈ E (S/μ) , then, by Lemma 2.5(3), μ_e ★ μ_a = μ_ea = μ_g for some g ∈ E (S) . Hence ea ∈ E (S) since μ is idempotent-pure, and so a ∈ E (S) since S is E-unitary. Thus μ_a ∈ E (S/μ) . This gives that S/μ is left E-unitary. Similarly, we can prove that S/μ is right E-unitary.

(3) Necessity. Let e ∈ E (S) , x, y ∈ S be such that exy ∈ E (S) . Then μ_exy ∈ E (S/μ) . That is, μ_e ★ μ_x ★ μ_y ∈ E (S/μ) . Note that S/μ is E-reflexive with respect to operation “★.” We have μ_e ★ μ_y ★ μ_x ∈ E (S/μ) . That is, μ_eyx ∈ E (S/μ) . By Lemma 2.5(3), μ_eyx = μ_f for some f ∈ E (S) . Hence eyx ∈ E (S) since μ is idempotent-pure, and so S is E-reflexive.

Sufficiency. Let μ_e ∈ E (S/μ) , μ_x, μ_y ∈ S/μ with μ_e ★ μ_x ★ μ_y ∈ E (S/μ) , where e ∈ E (S) . That is, μ_exy ∈ E (S/μ) . By Lemma 2.5(3), μ_exy = μ_g for some g ∈ E (S) . Hence exy ∈ E (S) since μ is idempotent-pure. Note that S is E-reflexive. We have eyx ∈ E (S) . Hence μ_eyx ∈ E (S/μ) . That is, μ_e ★ μ_y ★ μ_x ∈ E (S/μ) . Therefore, S/μ is E-reflexive for operation “★.”

(4) Suppose that μ is cancellative. Then, by Lemma 2.5(4), μ_e = μ_f for all e, f ∈ E (S) . Let μ_e ∈ E (S/μ) , μ_a ∈ S/μ be such that μ_e ★ μ_a ∈ E (S/μ) , where e ∈ E (S) . Then μ_e ★ μ_a = μ_{a
⁺} ★ μ_a = μ_a ∈ E (S/μ) . This gives that S/μ is left E-unitary. Similarly, we can prove that S/μ is right E-unitary. Therefore, S/μ is E-unitary with respect to operation “★.”

(5) Obviously, σ is an admissible congruence on S. By Lemma 2.5(1), we see that C_σ is a fuzzy admissible congruence on S . Next, we prove that C_σ is a fuzzy cancellative congruence on S . For this, let (C_σ) _a, (C_σ) _b, (C_σ) _c ∈ S/C_σ be such that (C_σ) _a ★ (C_σ) _b = (C_σ) _a ★ (C_σ) _c, that is, (C_σ) _ab = (C_σ) _ac. Then, by Lemma 2.4, we have C_σ (ab, ac) =1, that is, (ab, ac) ∈ σ . But σ is a cancellative congruence on S, so (b, c) ∈ σ . Thus C_σ (b, c) =1 . By Lemma 2.4, we get that (C_σ) _b = (C_σ) _c . This means that C_σ is a fuzzy left cancellative congruence on S . Similarly, we have that C_σ is a fuzzy right cancellative congruence on S . Therefore, by (4), S/C_σ is E-unitary for operation “★.” □

Theorem 4.3. Let μ be a fuzzy admissible congruence on a type B semigroup S. If S/μ is primitive for operation “★”, then S/μ is E-unitary for operation “★.”

Proof Let μ_e ∈ E (S/μ) , μ_a ∈ S/μ be such that μ_e ★ μ_a ∈ E (S/μ) , where e ∈ E (S) . Note that μ_e ★ μ_arx (S/μ) μ_e ★ μ_{a
⁺} since μ is a fuzzy admissible congruence and rx is a left congruence. Hence μ_e ★ μ_a = μ_e ★ μ_{a
⁺} since S/μ is rx-unipotent. On the other hand, μ_e ★ μ_{a
⁺} ≤ μ_{a
⁺} . We have μ_e ★ μ_{a
⁺} = μ_{a
⁺} since S/μ is primitive. Thus $μ_{a} = μ_{a^{+} a} = μ_{a^{+}} ★ μ_{a} = μ_{e} ★ μ_{a^{+}} ★ μ_{a} = μ_{e} ★ μ_{a} \in E (S / μ) .$ Dually, we can prove that μ_a ★ μ_e ∈ E (S/μ) implies μ_a ∈ E (S/μ) . Therefore, S/μ is E-unitary for operation “★.” □

Corollary 4.4. Let μ be an idempotent-separating fuzzy admissible congruence on a primitive type B semigroup S. Then S/μ is E-unitary for operation “★.”

Proof. It follows from Theorems 4.2(1) and 4.3. □

Remark 4.2. We remark here that S/μ is a primitive type B semigroup implies S/μ is E-unitary for an arbitrary idempotent-separating fuzzy admissible congruence μ on a type B semigroup S, but the converse is not true. In fact, in our Example 3.1, we have E (S) = {(m, m) | m ∈ N} and E (S/C_η) = {(C_η) _(m, m) | m ∈ N} . Let (C_η) _(m, m) ∈ E (S/C_η) , (C_η) _(p, q) ∈ S/C_η such that (C_η) _(m, m) * (C_η) _(p, q) ∈ E (S/C_η) . Then (C_η) _{(m, m)•(p, q)} = (C_η) _(t, q-p+t) ∈ E (S/C_η) , where t = max {m, p} . Hence, t = q - p + t, that is, q = p . Thus (C_η) _(p, q) = (C_η) _(p, p) ∈ E (S/C_η) . That is, S/C_η is right E-unitary. Similarly, we can prove that S/C_η is left E-unitary. However, in our Remark 4.1, we have prove that S/C_η is not primitive.

Theorem 4.5. Let μ be a fuzzy admissible congruence on a fundamental type B semigroup S. Then the following statements are true:

if μ is idempotent-separating, then S/μ is fundamental for operation “★ ″ ;

S/C_η is fundamental for operation “★.”

Proof. (1) For convenience, η^S/μ [resp., $η_{l}^{S / μ}, η_{r}^{S / μ}$ ] denotes η [resp., η_l, η_r] on S/μ, and η^S [resp., $η_{l}^{S}, η_{r}^{S}$ ] denotes η [resp., η_l, η_r] on S .

Let μ_a, μ_b ∈ S/μ be such that (μ_a, μ_b) ∈ η^S/μ . Then $(μ_{a}, μ_{b}) \in η_{l}^{S / μ} .$ Hence, (∀ e ∈ E (S)) (μ_e ★ μ_a) ^* = (μ_e ★ μ_b) ^*, and so μ_(ea)^* = (μ_ea) ^* = (μ_eb) ^* = μ_(eb)^* . Thus, (ea) ^* = (eb) ^* since μ is idempotent-separating. This gives that $(a, b) \in η_{l}^{S} .$ Similarly, we can prove that $(a, b) \in η_{r}^{S} .$ Therefore, (a, b) ∈ η^S . But S is fundamental. We have a = b, and so μ_a = μ_b . That is, S/μ is fundamental.

(2)By Lemma 2.5(2), it is easy to see that C_η is an idempotent-separating fuzzy admissible congruence on S. Thus, by (1), S/C_η is fundamental. □

5 Some applications

As in [27], for an arbitrary inverse semigroup S, we have that θ = {(a, b) | (∀ e ∈ E (S)) a^-1ea = b^-1eb} is the greatest idempotent-separating congruence on S contained in hc, where hc is an intersection of lc and rc . It is well known that an arbitrary inverse semigroup is type B, and that an arbitrary congruence on an inverse semigroup is an admissible congruence. In fact, we have that an arbitrary fuzzy congruence on an inverse semigroup is a fuzzy admissible congruence. Therefore, as applications of Theorem 4.2 and Corollary 4.4, the following corollaries hold.

Corollary 5.1. Let S be an inverse semigroup. Then

S is primitive if and only if S/C_θ is primitive for operation “★”;

if S is primitive, then S/C_θ is E-unitary for operation “★”.

Proof. (1) It follows directly from Theorem 4.2(1).

(2) It follows immediately from Corollary 4.4. □

Corollary 5.2. Let S be an arbitrary inverse semigroup and μ be an idempotent-separating fuzzy congruence on S. Then S is primitive if and only if S/μ is primitive for operation “★”.

Proof. It follows directly from Theorem 4.2(1). □

Corollary 5.3. Let S be an arbitrary inverse semigroup and μ be an idempotent-pure fuzzy congruence on S. Then the following statements are true:

S is E-unitary if and only if S/μ is E-unitary for operation “★”;

S is E-reflexive if and only if S/μ is E-reflexive for operation “★”.

Proof. (1) It follows immediately from Theorem 4.2(2).

(2) It follows directly from Theorem 4.2(3). □

We now give an example of an inverse semigroup which satisfies the conditions in Corollary 5.3 (1) and (2).

Example 5.1. Let S be the three-element semigroup whose multiplication table is given below:

	e	f	1
e	e	e	e
f	e	f	f
1	e	f	1

Then it is readily verified that S is an inverse semigroup, and that S is both E-unitary and E-reflexive. Suppose that μ is an fuzzy congruence on S. Then one checks readily that μ is idempotent-pure on S, and that S/μ is an inverse semigroup. In particular, it is easy to see that S/μ is both E-unitary and E-reflexive for operation “★”.

Next, we give an example of an inverse semigroup which satisfies the condition of Corollary 5.3 (2). But it does not satisfy the condition of Corollary 5.3 (1).

Example 5.2. Let S be the four-element semigroup with a multiplication table as follows:

	e	f	1	a
e	e	e	e	e
f	e	f	f	f
1	e	f	1	a
a	e	f	a	1

Then it is evident that S is an E-reflexive inverse semigroup with E (S) = {e, f, 1} . However, S is not E-unitary since e, ea ∈ E (S) and a ∉ E (S).

Now, Assume that μ is an idempotent-pure fuzzy congruence on S. Then it is easy to check that S/μ is an inverse semigroup with E (S/μ) = {μ_e, μ_f, μ₁} . Obviously, μ_x ★ μ_y ★ μ_z = μ_x ★ μ_z ★ μ_y ∈ E (S/μ) for all x, y, z ∈ E (S) . On the other hand, μ_e ★ μ_a ★ μ_f = μ_e ★ μ_f ★ μ_a = μ_f ★ μ_a ★ μ_e = μ_f ★ μ_e ★ μ_a = μ_e ∈ E (S/μ) , μ_e ★ μ_a ★ μ₁ = μ_e ★ μ₁ ★ μ_a = μ₁ ★ μ_a ★ μ_e = μ₁ ★ μ_e ★ μ_a = μ_e ∈ E (S/μ) and μ_f ★ μ_a ★ μ₁ = μ_f ★ μ₁ ★ μ_a = μ₁ ★ μ_a ★ μ_f = μ₁ ★ μ_f ★ μ_a = μ_f ∈ E (S/μ) .

Therefore, S/μ is an E-reflexive inverse semigroup for operation ★.

Next, we show that S/μ is not E-unitary. To see it, we only prove that μ_a ∉ E (S/μ) since μ_e ★ μ_a = μ_e ∈ E (S/μ) . Suppose that μ_a ∈ E (S/μ) . Then μ_a = μ_x for some x ∈ E (S) . Note that μ is an idempotent-pure fuzzy congruence on S. We have a = x for some x ∈ E (S) . This contradiction suggests that S/μ is not E-unitary.

6 Conclusions

It is well known that one of the areas of semigroup theory which showed great popularity almost from its beginning is that of inverse semigroups. The class of E-unitary semigroups is one of the most important inverse semigroup theory. There are two main reasons for this: firstly, every inverse semigroup is an idempotent-separating homomorphic image of an E-unitary semigroup; and secondly, many naturally occurring inverse semigroups are E-unitary. In this paper, we study some classes of type B semigroups which are generalized inverse semigroups by using the notion of fuzzy admissible congruence of an abundant semigroup. It is a new way to depicting the properties and characterizations of semigroups. In particular, we consider fuzzy congruences on type B semigroups with E-properties. It is an interesting thing since our these results can be applied to inverse semigroups with E-properties (for example, E-unitary, E-reflexive) and cancellative monoids. In fact, our result can also employ other generalized inverse semigroup theory.

Footnotes

Acknowledgments

This work is supported by the NNSF(CN) (Nos. 11261018; 11961026), the NSF of Jiangxi Province (No. 20224BAB211005), the JiangXi Educational Department Natural Science Foundation of China (No. GJJ2200634) and the Science and Technology Foundation of Guizhou Province (No. QIANKEHEJICHU-ZK[2021]313).

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