# «0-TIGHT COMPLETELY 0-SIMPLE SEMIGROUPS BY HSING Y. WU Abstract A semigroup is 0-tight if each of its congruences is uniquely determined by each of ...»

A compleltely 0-simple semigroup M0 [G; I, Λ; P ] is 0tight if and only if the rectangular 0-band (I × Λ) ∪ {0} whose multiplication is in terms of P with entries in G0 is 0-tight.

Proof. Let S = M0 [G; I, Λ; P ], and let Y = (I × Λ) ∪ {0}. Suppose that Y is 0-tight. By Theorem 2.2 there are cases of equivalences EI and EΛ. We only verify the case where EI = 1I and EΛ = 1Λ. In this case, S = 1I and T = 1Λ.

Let ρ be a proper congruence on S. To show that ρ is uniquely determined by each of its congruence classes that do not contain zero, it suﬃces to verify

**the following:**

(i, a, λ) ρ = {(i, ga, λ) : g ∈ N }.

To show the converse, if g ∈ N, then (pξi )(ga)a−1 (pξi )−1 = (pξi )g(pξi )−1 in N for every ξ ∈ Λ such that pξi = 0. Hence (i, a, λ) ρ (i, b, λ).

Notice that now every congruence on S is uniquely determined by each of the congruence classes that do not contain zero. So S is a 0-tight semigroup.

Conversely, suppose S is 0-tight. Since Y is a homomorphic image of S, Y is 0-tight.

Unlike completely 0-simple semigroups, we do not need equivalences EI and EΛ to investigate congruences on completely simple semigroups. We only have to discuss the equivalence relations S and T on I and Λ, respectively (see p. 90 in [2]). This makes our classiﬁcation a lot easier.

To classify ﬁnite tight rectangular bands, we consider various partitions of m and n. There is a speciﬁc diﬀerence between the classiﬁcation of ﬁnite rectangular 0-bands and that of ﬁnite rectangular bands. Let S be a ﬁnite rectangular band. We adopt similar terminology in Section 2.

Again there is a duality between ﬁnite sets I and Λ. As proved in Theorem 1.1.3 in [2], a semigroup is a rectangular band if and only if it is isomorphic to the direct product of a left zero semigroup A and a right zero semigroup B. The proof of Corollary 3.2 follows the proof of Theorem 2.2.

**Corollary 3.2. Let S be a rectangular band. Then the following statements are equivalent:**

(1) S is tight.

(2) 1S is uniquely determined by each of its congruence classes.

(3) |A| ≤ 2 and |B| ≤ 2.

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8. Hsing Y. Wu, Tight congruences on semigroups, Doctoral Dissertation, Univ. of Arkansas, Fayetteville, 2004, 11-18.

Chung Jen Junior College of Nursing, Health Science and Management, # 217, Hung Mao Pi, Jia Yi City, Taiwan 60077.

E-mail: m105@mail.cjns.cy.edu.tw