Abstract
We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.
Highlights
Introduction and PreliminariesThe term flow monoid first arose in unpublished typescript of Chase; in this manuscript, Chase determines the structure of the flow monoid and its group of units in general category theory which has categories with fixed vertices set X
We describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands
We find the general structure of the flow monoids on Cauchy categories
Summary
The term flow monoid first arose in unpublished typescript of Chase; in this manuscript, Chase determines the structure of the flow monoid (named it in his paper incidence monoid) and its group of units in general category theory which has categories with fixed vertices set X. We describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. A flow on a category C with vertex set X is a function φ : X → C that is a section to the source map: that is, for all x ∈ X, (xφ)σ = x, let Φ(C) denote the set of all flows on C; Φ(C) is a monoid, with composition ∗. In the Cauchy category (S), we have exactly one arrow from e to f (for all e, f ∈ E(S)) if and only if S is a rectangular band
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