MORITA EQUIVALENCE OF SEMIGROUPS REVISITED: FIRM SEMIGROUPS

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1 MORITA EQUIVALENCE OF SEMIGROUPS REVISITED: FIRM SEMIGROUPS László Márki Rényi Institute, Budapest joint work with Valdis Laan and Ülo Reimaa Berlin, 10 October 2017

2 Monoids Two monoids S and T are Morita equivalent if there exist elements u, v, w T such that u 2 = u, vu = v, uw = w, vw = 1 and utu = S.

3 Monoids Two monoids S and T are Morita equivalent if there exist elements u, v, w T such that u 2 = u, vu = v, uw = w, vw = 1 and utu = S. Corollary 1. Morita equivalence of two monoids very often reduces to isomorphism.

4 Monoids Two monoids S and T are Morita equivalent if there exist elements u, v, w T such that u 2 = u, vu = v, uw = w, vw = 1 and utu = S. Corollary 1. Morita equivalence of two monoids very often reduces to isomorphism. Corollary 2. The categories of all acts over two arbitrary semigroups are equivalent if and only if the two semigroups are isomorphic.

5 Semigroups, acts 1 factorisable semigroup: every element can be written as a product

6 Semigroups, acts 1 factorisable semigroup: every element can be written as a product semigroup S with weak local units: s S u, v S su = s = vs

7 Semigroups, acts 1 factorisable semigroup: every element can be written as a product semigroup S with weak local units: s S u, v S su = s = vs semigroup S with local units: u and v can be chosen to be idempotent

8 Semigroups, acts 1 factorisable semigroup: every element can be written as a product semigroup S with weak local units: s S u, v S su = s = vs semigroup S with local units: u and v can be chosen to be idempotent A S unitary act: AS = A

9 Semigroups, acts 1 factorisable semigroup: every element can be written as a product semigroup S with weak local units: s S u, v S su = s = vs semigroup S with local units: u and v can be chosen to be idempotent A S unitary act: AS = A A S firm act: the mapping is bijective µ A : A S A, a s as

10 Semigroups, acts 1 factorisable semigroup: every element can be written as a product semigroup S with weak local units: s S u, v S su = s = vs semigroup S with local units: u and v can be chosen to be idempotent A S unitary act: AS = A A S firm act: the mapping is bijective µ A : A S A, a s as A S unitary µ A surjective, hence firm acts are unitary.

11 Semigroups, acts 1 factorisable semigroup: every element can be written as a product semigroup S with weak local units: s S u, v S su = s = vs semigroup S with local units: u and v can be chosen to be idempotent A S unitary act: AS = A A S firm act: the mapping is bijective µ A : A S A, a s as A S unitary µ A surjective, hence firm acts are unitary. S factorisable µ S surjective

12 Semigroups, acts 2 S is a firm semigroup: S S (or S S) is a firm act

13 Semigroups, acts 2 S is a firm semigroup: S S (or S S) is a firm act S is a right fair semigroup: every subact of a unitary right S-act is unitary.

14 Semigroups, acts 2 S is a firm semigroup: S S (or S S) is a firm act S is a right fair semigroup: every subact of a unitary right S-act is unitary. S is a fair semigroup: S is left and right fair.

15 Semigroups, acts 2 S is a firm semigroup: S S (or S S) is a firm act S is a right fair semigroup: every subact of a unitary right S-act is unitary. S is a fair semigroup: S is left and right fair. A S s-unital act: for every a A there exists s S such that as = a.

16 Semigroups, acts 2 S is a firm semigroup: S S (or S S) is a firm act S is a right fair semigroup: every subact of a unitary right S-act is unitary. S is a fair semigroup: S is left and right fair. A S s-unital act: for every a A there exists s S such that as = a. S is a right fair semigroup every unitary right S-act is s-unital.

17 Semigroups, acts 2 S is a firm semigroup: S S (or S S) is a firm act S is a right fair semigroup: every subact of a unitary right S-act is unitary. S is a fair semigroup: S is left and right fair. A S s-unital act: for every a A there exists s S such that as = a. S is a right fair semigroup every unitary right S-act is s-unital. Fair semigroups are counterparts of xst-rings considered by García and Marín, based on work of Xu, Shum, and Turner-Smith.

18 Semigroups, acts 2 S is a firm semigroup: S S (or S S) is a firm act S is a right fair semigroup: every subact of a unitary right S-act is unitary. S is a fair semigroup: S is left and right fair. A S s-unital act: for every a A there exists s S such that as = a. S is a right fair semigroup every unitary right S-act is s-unital. Fair semigroups are counterparts of xst-rings considered by García and Marín, based on work of Xu, Shum, and Turner-Smith. A S non-singular act: if as = a s for all s S, then a = a (a, a A)

19 Categories of acts Act S ( S Act, S Act T ) all right S-acts (left S-acts, (S, T )-biacts)

20 Categories of acts Act S ( S Act, S Act T ) all right S-acts (left S-acts, (S, T )-biacts) UAct S all unitary right S-acts

21 Categories of acts Act S ( S Act, S Act T ) all right S-acts (left S-acts, (S, T )-biacts) UAct S all unitary right S-acts FAct S all firm right S-acts

22 Categories of acts Act S ( S Act, S Act T ) all right S-acts (left S-acts, (S, T )-biacts) UAct S all unitary right S-acts FAct S all firm right S-acts NAct S all non-singular right S-acts

23 Categories of acts Act S ( S Act, S Act T ) all right S-acts (left S-acts, (S, T )-biacts) UAct S all unitary right S-acts FAct S all firm right S-acts NAct S all non-singular right S-acts CAct S those A S acts for which the mapping λ S : S Act S (S, A) : λ S (a)(s) = as is an isomorphism

24 Categories of acts Act S ( S Act, S Act T ) all right S-acts (left S-acts, (S, T )-biacts) UAct S all unitary right S-acts FAct S all firm right S-acts NAct S all non-singular right S-acts CAct S those A S acts for which the mapping λ S : S Act S (S, A) : λ S (a)(s) = as is an isomorphism Fx Act S all fixed acts: those acts A S for which the mapping S SAct(S, A) A : s f (s)f is an isomorphism

25 Adjoint situations, equivalences 1 Let S be a firm semigroup. Act S = Act S S I UAct S S Act S (S, ) I I S Act S (S, ) S Act S (S, )S I CAct S S Act S (S, ) FAct S Act(S, )S S NAct S

26 Adjoint situations, equivalences 1 Let S be a firm semigroup. Act S = Act S S I UAct S S Act S (S, ) I I S Act S (S, ) S Act S (S, )S I CAct S S Act S (S, ) FAct S Act(S, )S S NAct S The equivalence between CAct S and NAct S can be given as S CAct S Act S (S, ) NAct S.

27 Adjoint situations, equivalences 2 FAct S is an essential colocalization of UAct S with coreflection S; NAct S is an essential localization of UAct S with reflection Act S (S, )S.

28 Adjoint situations, equivalences 2 FAct S is an essential colocalization of UAct S with coreflection S; NAct S is an essential localization of UAct S with reflection Act S (S, )S. Over a firm semigroup, firm acts are the same as fixed acts.

29 Equivalence functors between categories of firm acts Let S and T be firm semigroups and S P T be a biact such that P T is firm. Then the functor P : FAct S FAct T is left adjoint to the functor Act T (P, ) S : FAct T FAct S.

30 Equivalence functors between categories of firm acts Let S and T be firm semigroups and S P T be a biact such that P T is firm. Then the functor P : FAct S FAct T is left adjoint to the functor Act T (P, ) S : FAct T FAct S. Let S and T be firm semigroups and F : FAct S FAct T and G : FAct S FAct T be mutually inverse equivalence functors. Then F = Act S (G(T ), ) T a G = Act T (F (S), ) S.

31 Equivalence functors between categories of firm acts Let S and T be firm semigroups and S P T be a biact such that P T is firm. Then the functor P : FAct S FAct T is left adjoint to the functor Act T (P, ) S : FAct T FAct S. Let S and T be firm semigroups and F : FAct S FAct T and G : FAct S FAct T be mutually inverse equivalence functors. Then F = Act S (G(T ), ) T a G = Act T (F (S), ) S. Let S and T be firm semigroups and F : FAct S FAct T and G : FAct S FAct T be mutually inverse equivalence functors. Then F = F (S), G = G(T ). Moreover, the left acts S F (S) and T G(T ) are firm.

32 Morita contexts a sixtuple (S, T, S P T, T Q S, θ, φ), where S and T are semigroups, SP T S Act T ands T Q S T Act S are biacts, and θ : S (P Q) S S S S, φ : T (Q P) T T T T are biact morphisms such that, for every p, p P and q, q Q, θ(p q)p = pφ(q p ), qθ(p q ) = φ(q p)q.

33 Morita contexts a sixtuple (S, T, S P T, T Q S, θ, φ), where S and T are semigroups, SP T S Act T ands T Q S T Act S are biacts, and θ : S (P Q) S S S S, φ : T (Q P) T T T T are biact morphisms such that, for every p, p P and q, q Q, θ(p q)p = pφ(q p ), qθ(p q ) = φ(q p)q. A Morita context is unitary if S P T and T Q S are unitary biacts, surjective if θ and φ are surjective, bijective if θ and φ are bijective.

34 Morita equivalence and strong Morita equivalence 1 Semigroups S and T are (right) Morita equivalent if the categories FAct S and FAct T are equivalent.

35 Morita equivalence and strong Morita equivalence 1 Semigroups S and T are (right) Morita equivalent if the categories FAct S and FAct T are equivalent. Semigroups S and T are strongly Morita equivalent, if they are contained in a unitary surjective Morita context.

36 Morita equivalence and strong Morita equivalence 1 Semigroups S and T are (right) Morita equivalent if the categories FAct S and FAct T are equivalent. Semigroups S and T are strongly Morita equivalent, if they are contained in a unitary surjective Morita context. Earlier results:

37 Morita equivalence and strong Morita equivalence 1 Semigroups S and T are (right) Morita equivalent if the categories FAct S and FAct T are equivalent. Semigroups S and T are strongly Morita equivalent, if they are contained in a unitary surjective Morita context. Earlier results: If S is strongly Morita equivalent to any semigroup, including itself, then S is factorisable.

38 Morita equivalence and strong Morita equivalence 1 Semigroups S and T are (right) Morita equivalent if the categories FAct S and FAct T are equivalent. Semigroups S and T are strongly Morita equivalent, if they are contained in a unitary surjective Morita context. Earlier results: If S is strongly Morita equivalent to any semigroup, including itself, then S is factorisable. Lawson (Right) Morita equivalence and strong Morita equivalence coincide for semigroups with local units.

39 Morita equivalence and strong Morita equivalence 2 Chen Shum For arbitrary factorisable semigroups S és T, the categories NAct S and NAct T are equivalent if and only if the semigroups S/ζ S and T /ζ T are strongly Morita equivalent, where the congruence ζ A is defined, for an act A S, by ζ A = {(a 1, a 2 ) A 2 a 1 s = a 2 s for all s S}.

40 Morita equivalence and strong Morita equivalence 2 Chen Shum For arbitrary factorisable semigroups S és T, the categories NAct S and NAct T are equivalent if and only if the semigroups S/ζ S and T /ζ T are strongly Morita equivalent, where the congruence ζ A is defined, for an act A S, by ζ A = {(a 1, a 2 ) A 2 a 1 s = a 2 s for all s S}. Laan Márki Let S and T be fair semigroups such that U(S) and U(T ) have common weak local units, where U(S) = {s S s = us = sv for some u, v S} (this is an ideal in S). Then S and T are right Morita equivalent if and only if U(S) and U(T ) are strongly Morita equivalent.

41 Morita equivalence need not be strong non-factorisable example S is a non-trivial semigroup with zero multiplication: then S is fair and U(S) = {0} has common weak local units. S is right and left Morita equivalent to the one-element semigroup but it is not factorisable.

42 Main theorem For firm semigroups S and T, the following conditions are equivalent: 1 The categories FAct S and FAct T are equivalent.

43 Main theorem For firm semigroups S and T, the following conditions are equivalent: 1 The categories FAct S and FAct T are equivalent. 2 The categories S FAct and T FAct are equivalent.

44 Main theorem For firm semigroups S and T, the following conditions are equivalent: 1 The categories FAct S and FAct T are equivalent. 2 The categories S FAct and T FAct are equivalent. 3 There exists a unitary bijective Morita context containing S and T.

45 Main theorem For firm semigroups S and T, the following conditions are equivalent: 1 The categories FAct S and FAct T are equivalent. 2 The categories S FAct and T FAct are equivalent. 3 There exists a unitary bijective Morita context containing S and T. 4 There exists a unitary surjective Morita context containing S and T.

46 Main theorem For firm semigroups S and T, the following conditions are equivalent: 1 The categories FAct S and FAct T are equivalent. 2 The categories S FAct and T FAct are equivalent. 3 There exists a unitary bijective Morita context containing S and T. 4 There exists a unitary surjective Morita context containing S and T. 5 There exists a surjective Morita context containing S and T.