Constructing inverse monoids from small categories
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 \textit{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.
scientific article; zbMATH DE number 4031932