Abstract
Two partially ordered monoids S and T are called Morita equivalent if the categories of right S-posets and right T -posets are Pos-equivalent as categories enriched over the category Pos of posets. We give a description of Pos-prodense biposets and prove Morita theorems I, II, and III for partially ordered monoids.
Highlights
At the beginning of the 1970s, Knauer [5] and Banaschewski [2] proved the first fundamental results about Morita equivalence of monoids, establishing a theory parallel to the classical theory of Morita equivalent rings
Two partially ordered monoids S and T are called Morita equivalent if the categories of right S-posets and right T -posets are Pos-equivalent as categories enriched over the category Pos of posets
The aim of this paper is to develop a theory of Morita equivalent partially ordered monoids
Summary
At the beginning of the 1970s, Knauer [5] and Banaschewski [2] proved the first fundamental results about Morita equivalence of monoids, establishing a theory parallel to the classical theory of Morita equivalent rings (see [1] for an overview about that). Left S-posets, (T, S)-biposets), where the morphisms are order and monoid action preserving mappings. For fixed elements s ∈ S, t ∈ T , and SAT ∈ SPosT , the mappings ρt : A → A, a → a ·t, and λs : A → A, a → s · a, are morphisms in SPos and PosT , respectively. Pomonoids S and T are called Morita equivalent if the categories PosS and PosT are Posequivalent. An S-poset AS is a cyclic projective generator in PosS if and only if AS ∼= eSS for an idempotent e ∈ S with eJ 1. We call a biposet SAT faithfully balanced if the pomonoid homomorphisms λ : S → End(AT ) and ρ : T → End(SA) are isomorphisms. If AT is a cyclic projective generator and λ : S → End(AT ) is an isomorphism SAT is faithfully balanced
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