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Constructing inverse monoids from small categories. (English) Zbl 0634.18002

The concept of a division category D(S) for a given semigroup S was introduced by the author [Mem. Am. Math. Soc. 157, 1-66 (1975; Zbl 0303.20048)], and a variation was developed by M. Loganathan [J. Algebra 70, 375-393 (1981; Zbl 0465.20063)]. The first section of this paper begins by defining an abstract division category (ADC) to be a pair (D,I), where D is a small category having finite pushouts all of whose morphisms are epics, and I is a distinguished object such that Hom(I,X) never vanishes for any object in D. The main result in this section is Theorem 1.15 asserting that Loganathan pairs always yield an ADC. The second section presents the main construction of this work: for each pair (D,I) an inverse monoid of fractions, S[D,I], is constructed. Theorem 2.2 is the main result in this section and asserts that S[D,I] is an inverse monoid. A second result, Theorem 2.5, asserts that equivalence of two pairs, (D,I)\(\cong (D',I')\), induces isomorphism of the derived monoids, S[D,I]\(\cong S[D',I']\). In section 3 it is proved that there exists a canonical isomorphism \(\Theta\) : \(S\cong S[D(s),1]\) for a given inverse monoid S, and that two inverse monoids are isomorphic iff their induced division categories are pair equivalent. Theorem 4.4 and Corollary 4.5 provide noncanonial equivalences of (D,I) with (D(S),1), where \(S=S[D,I]\) and D(S) is the Loganathan category of S. In the fifth section the functor \(\Sigma_ 0: D\to INV\) from the category of abstract division categories to the category of inverse monoids is studied. The last section contains a brief study of (dual) symmetrical inverse monoids in full generality.
Reviewer: E.Katsov

MSC:

18B99 Special categories
20M50 Connections of semigroups with homological algebra and category theory
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References:

[1] Aznar, E. and A. Sevilla,Equivalences between D-categories of inverse monoids, Semigroup Forum, to appear. · Zbl 0957.20050
[2] Clifford, A.,A class of d-simple semigroups, Amer. J. Math., 75(1953), 547–556. · Zbl 0051.01302
[3] Clifford, A. and G. Preston,The Algebraic Theory of Semigroups, II, AMS Surveys, 7(1967), 101–108.
[4] Leech, J.,H-coextensions of monoids, Memoirs AMS, 157(1975), 1–66. · Zbl 0303.20048
[5] Leech, J.,The D-category of a semigroup, Semigroup Forum, 11(1976), 283–296. · Zbl 0338.20094
[6] Leech, J.,The D-category of a monoid, Semigroup Forum, 34(1986), 89–116. · Zbl 0612.20035
[7] Loganathan, M.,Cohomology of inverse semigroups Journal of Algebra, 70(1981), 375–393. · Zbl 0465.20063
[8] Loganathan, M.,Idempotent-separating extensions of regular semigroups with abelian kernel, J. Australian Math. Soc., 32(1982), 104–113. · Zbl 0485.20053
[9] Loganathan, M.,Cohomology and extensions of regular semigroup, J. Australian Math. Soc., 35(1983), 178–193. · Zbl 0524.20046
[10] Mitchell, B.,Theory of Categories, Academic Press, New York, 1965. · Zbl 0136.00604
[11] Petrich, M.,Inverse Semigroups, Wiley-Interscience, New York, 1984. · Zbl 0546.20053
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