Morita equivalence of semigroups revisited: Firm semigroups

Abstract

We define firm semigroups and firm acts as non-additive analogues of firm rings and firm modules. Using the categories of firm acts we develop Morita theory for firm semigroups. We show that equivalence functors between categories of firm acts over two firm semigroups have to be tensor multiplication functors. Our main result states that the categories of firm right acts over two firm semigroups are equivalent if and only if these semigroups are strongly Morita equivalent, which means that they are contained in a unitary Morita context with bijective mappings. We also investigate other categories of acts which have been used earlier to develop Morita equivalence. The main tool in our work is adjoint functors. We prove that over firm semigroups all the considered categories are equivalent to the category of firm acts. All this suggests that firm semigroups and firm acts are the natural environment to study Morita equivalence of semigroups.

Perfection for semigroups

“…As in Proposition 3.16 of [18], one can prove that and are equivalence functors inverse to each other (see also Theorem 5.9 in [14]). We will prove that takes cyclic acts to cyclic acts.…”

“…Equivalently, by Proposition 4.9 in [16], and are Morita equivalent firm semigroups. According to Theorem 5.9 in [14], there exists a unitary Morita context with bijective mappings connecting and . Consequently,…”

On connections between Morita semigroups and strong Morita equivalence

“…A semigroup S is said to have common weak local units if for every s, s ′ ∈ S there exist u, v ∈ S such that us = s, us ′ = s ′ and sv = s, s ′ v = s ′ . Semigroups with common weak local units are introduced in [9] and also shown to be firm (cf. Proposition 2.4).…”

“…Let S be factorizable such that it is not firm. An example of such a semigroup can be found in [9]. Then S ⊗ S S is firm both as a biact and a semigroup by Theorem 2.6 in [10] and µ : S ⊗ S S / / S, s ⊗ s ′ → ss ′ , is a biact morphism in the Morita context connecting S and S ⊗ S S (cf.…”

“…⇒ 2. assume the firm semigroups S and T are strongly Morita equivalent. By Theorem 5.9 in [9], they are connected by a unitary Morita context (S, T, S P T , T Q S , θ, φ) with bijective mappings. Then P ⊗ T Q defined by φ is a unitary Morita semigroup over T .…”