Morita equivalence of factorizable semigroups

Laan, Valdis; Reimaa, Ülo

doi:10.1142/s0218196719500243

Cited by 8 publications

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“…[11]). However, in the subclass of factorizable semigroups, it is sufficient to consider strong Morita equivalence, which coincides with the category theoretical Morita equivalence by Theorem 4.11 in [10].…”

Section: Introductionmentioning

confidence: 92%

“…By Laan and Reimaa [10], we have that a semigroup S is factorizable if and only if S ⊗ S is a firm semigroup. In the same article it is shown that A ⊗ S B = A ⊗ S⊗S B holds for factorizable semigroups S. While we do not know, whether Theorem 2.12 holds for factorizable semigroups, we can conclude the following.…”

Section: If T Is a Semigroup With Common Weak Local Units Then All St...mentioning

confidence: 99%

“…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. Proposition 4.7 in [10]).…”

Section: If T Is a Semigroup With Common Weak Local Units Then All St...mentioning

confidence: 99%

“…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. Proposition 4.7 in [10]). The morphism µ is a strict local isomorphism by Theorem 2.8, but it cannot be injective, because S is not firm.…”

Section: If T Is a Semigroup With Common Weak Local Units Then All St...mentioning

confidence: 99%

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On connections between Morita semigroups and strong Morita equivalence

Lepik¹

2021

Preprint

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A surjective Morita context connecting semigroups S and T yields a Morita semigroup and a strict local isomorphism from it onto S along which idempotents lift. We describe strong Morita equivalence of firm semigroups in terms of Morita semigroups and isomorphisms. We also generalize some of Hotzel's theorems to semigroups with weak local units. In particular, the Morita semigroup induced by a dual pair β over a semigroup with weak local units is isomorphic to Σ β . 2010 Mathematics Subject Classification. 20M30.

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Section: Introductionmentioning

confidence: 92%

Section: If T Is a Semigroup With Common Weak Local Units Then All St...mentioning

confidence: 99%

Section: If T Is a Semigroup With Common Weak Local Units Then All St...mentioning

confidence: 99%

Section: If T Is a Semigroup With Common Weak Local Units Then All St...mentioning

confidence: 99%

See 2 more Smart Citations

On connections between Morita semigroups and strong Morita equivalence

Lepik¹

2021

Preprint

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“…By Proposition 4.9 in [16], these functors are inverse equivalence functors. We also recall that is considered as a semigroup with the multiplication , and the action of is .…”

Section: Morita Invariancementioning

confidence: 99%

Perfection for semigroups

Laan¹,

Lepik²

2023

Proceedings of the Edinburgh Mathematical Society

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We call a semigroup right perfect if every object in the category of unitary right acts over that semigroup has a projective cover. In this paper, we generalize results about right perfect monoids to the case of semigroups. In our main theorem, we will give nine conditions equivalent to right perfectness of a factorizable semigroup. We also prove that right perfectness is a Morita invariant for factorizable semigroups.

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

2020

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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.

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Morita equivalence of factorizable semigroups

Cited by 8 publications

References 7 publications

On connections between Morita semigroups and strong Morita equivalence

On connections between Morita semigroups and strong Morita equivalence

Perfection for semigroups

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

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