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A semigroup is called factorizable if each of its elements can be written as a product. We study equivalences and adjunctions between various categories of acts over a fixed factorizable semigroup. We prove that two factorizable semigroups are Morita equivalent if and only if they are strongly Morita equivalent. We also show that Morita equivalence of finite factorizable semigroups is algorithmically decidable in finite time.

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.

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We study properties of the lattice of unitary ideals of a semigroup. In particular, we show that it is a quantale. We prove that if two semigroups are connected by an acceptable Morita context then there is an isomorphism between the quantales of unitary ideals of these semigroups. Moreover, factorisable ideals corresponding to each other under this isomorphism are strongly Morita equivalent. Mathematics Subject Classification. 20M30, 20M50.24 Page 2 of 14 V. Laan, L. Márki andÜ. Reimaa Algebra Univers.assumptions on the semigroups and requiring only acceptability for the context that connects the two semigroups. In this paper, semigroups and ideals are allowed to be empty and S, T will always stand for semigroups.are biact morphisms such that, for every p, p ∈ P and q, q ∈ Q,This context is said to be surjective if θ and φ are surjective, and unitary if the biacts S P T and T Q S are unitary. Example 1.2. Let S be a semigroup and let E be the set of its idempotents. Then we have a unitary Morita context (S, ESE, S SE ESE , ESE ES S , θ, φ) where θ : SE ⊗ ES → S, se ⊗ e s → see s , φ : ES ⊗ SE → ESE, es ⊗ s e → ess e . If S = SES then this context is also surjective. Acceptable Morita contexts for semigroups were introduced in [1] as non-additive analogues of acceptable Morita contexts of rings (see [8]). The existence of an acceptable Morita context with an extra condition guarantees Morita equivalence of the semigroups in that context (see [1, Theorem 2.4]). While unitary surjective Morita contexts can exist only between factorisable semigroups, acceptable Morita contexts can also connect nonfactorisable semigroups (see Example 1.4 and Example 3.8) giving another possibility (besides the equivalence of categories) for developing Morita theory for arbitrary semigroups. Definition 1.3. A Morita context (S, T, S P T , T Q S , θ, φ) is said to be right acceptable if (1) for every sequence (s m ) m∈N ∈ S N , there exists m 0 ∈ N such that s m0 . . . s 2 s 1 ∈ im(θ), (2) for every sequence (t m ) m∈N ∈ T N , there exists m 0 ∈ N such that t m0 . . . t 2 t 1 ∈ im(φ). Left acceptable Morita contexts are defined dually. A Morita context is acceptable if it is both right and left acceptable. Clearly, every surjective Morita context is acceptable. The converse does not hold. Vol. 81 (2020) Ideals of semigroups and Morita contexts Page 3 of 14 24 Example 1.4. Let n ≥ 2, S = s | s n = s n+1 and let T = {0} where 0 := s n . Then T is an ideal of the semigroup S and we have a Morita context with homomorphismswhere φ is surjective, but θ is not. However, this context is acceptable.