Lattices and quantales of ideals of semigroups and their preservation under Morita contexts

Abstract

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.

Morita contexts and unitary ideals of rings

“…In several papers ([2, Proposition 3.3], [7,Proposition 3.5], [3,Theorem 3.3]) it has been shown that lattices of certain ideals of Morita equivalent rings (of some type) are isomorphic. In this paper, taking inspiration from the recent developments in semigroup theory [10], we study under which conditions on a Morita context the quantales of unitary ideals are isomorphic.…”