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«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 ...»

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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.

–  –  –

1. A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. I (1961), Math. Surveys of the Amer. Math. Soc., 7, Providence.

2. J. M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995.

3. G. Lallement, A note on congruences on Rees matrix semigroups, Semigroup Forum, 8(1974), 89-92.

4. W. D. Munn, Embedding semigroups in congruence-free semigroups, Semigroup Forum, 4(1972), 46-60.

5. G. B. Preston, Congruences on completely 0-simple semigroups, Proc. London Math. Soc. (3), 11 (1961), 557-576.

6. D. Rees, On semi-groups, Math. Proc. Cambridge Philos. Soc., 36(1940), 387-400.

7. B. M. Schein, A remark concerning congruences on [0-] bisimple inverse semigroups, Semigroup Forum, 3(1971), 80-83.

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

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