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Morita equivalence of semigroups with local units. (English) Zbl 1229.20060
Let $S$ be a semigroup. A left $S$-act $X$ is called `closed’ if the map $\mu_X\colon S\otimes X\to X$ given by $\mu_X(s\otimes x)=sx$ is surjective and injective. The full subcategory (of all left $S$-acts and all $S$-homomorphisms) of all closed left $S$-acts is denoted by $S$-$\bold{FAct}$. Two semigroups $S$ and $T$ are called `Morita equivalent’ if the categories $S$-$\bold{FAct}$ and $T$-$\bold{FAct}$ are equivalent. A semigroup $S$ is called an `enlargement’ of its subsemigroup $U$ if $S=SUS$ and $U=USU$. A `consolidation’ of a strongly connected category $C$ is a map $q\colon\text{Obj}(C)\times\text{Obj}(C)\to\text{Mor}(C)$, $q(A,B)\colon B\to A$, such that $q(A,A)=1_A$. A small category $C$ equipped with a consolidation $q$ can be made into a semigroup $C^q$ by defining $x\circ y=xq(A,B)y$ where $x$ has domain $A$ and $y$ has codomain $B$. It is proved that semigroups with local units $S$ and $T$ are Morita equivalent if and only if one of the following conditions is satisfied: 1) the Cauchy completions $C(S)$ and $C(T)$ are equivalent; 2) $S$ and $T$ have a joint enlargement which can be chosen to be regular if $S$ and $T$ are both regular; 3) there is a unitary Morita context $(S,T,P,Q,\langle-,-\rangle,[-,-])$ with surjective mappings; 4) there is a consolidation $q$ on $C(S)$ and a local isomorphism $C(S)^q\to T$. Semigroups with local units Morita equivalent to a semigroup satisfying certain additional conditions (for example, to be a group, inverse semigroup, semilattice or orthodox semigroup) are described as well.

MSC:
20M10General structure theory of semigroups
20M50Connections of semigroups with homological algebra and category theory
18B40Groupoids, semigroupoids, semigroups, groups (viewed as categories)
20M17Regular semigroups
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References:
[1] Abrams, G. D.: Morita equivalence for rings with local units, Comm. algebra 11, 801-837 (1983) · Zbl 0503.16034 · doi:10.1080/00927878308822881
[2] Adámek, J.; Herrlich, H.; Strecker, G.: Abstract and concrete categories, (1990) · Zbl 0695.18001
[3] Banaschewski, B.: Functors into categories of M-sets, Abh. math. Sem. univ. Hamburg 38, 49-64 (1972) · Zbl 0257.18011 · doi:10.1007/BF02996922
[4] Chen, Y. Q.; Shum, K. P.: Morita equivalence for factorisable semigroups, Acta math. Sin. 17, 437-454 (2001) · Zbl 0991.20046 · doi:10.1007/s101140000056
[5] J. Funk, M.V. Lawson, B. Steinberg, Characterisations of Morita equivalence of inverse semigroups, Heriot--Watt University, 2009, Preprint. · Zbl 1229.20064
[6] Howie, J. M.: Fundamentals of semigroup theory, (1995) · Zbl 0835.20077
[7] Khan, T. A.; Lawson, M. V.: A characterisation of a class of semigroups with locally commuting idempotents, Periodica math. Hungar. 40, 85-107 (2000) · Zbl 0973.20053 · doi:10.1023/A:1010327307672
[8] Kilp, M.; Knauer, U.; Mikhalev, A. V.: Monoids, acts and categories, (2000) · Zbl 0945.20036
[9] Knauer, U.: Projectivity of acts and Morita equivalence of monoids, Semigroup forum 3, 359-370 (1972) · Zbl 0231.18013 · doi:10.1007/BF02572973
[10] V. Laan, L. Márki, On strong Morita equivalence of semigroups (in preparation).
[11] Lam, T. Y.: Lectures on rings and modules, (1999) · Zbl 0911.16001
[12] Lane, S. Mac: Categories for the working mathematician, (1998) · Zbl 0906.18001
[13] Lawson, M. V.: Enlargements of regular semigroups, Proc. edinb. Math. soc. 39, 425-460 (1996) · Zbl 0862.20047 · doi:10.1017/S001309150002321X
[14] Lawson, M. V.; Márki, L.: Enlargements and coverings by Rees matrix semigroups, Monatsh. math. 129, 191-195 (2000) · Zbl 0965.20037 · doi:10.1007/s006050050070
[15] Márki, L.; Steinfeld, O.: A Rees construction without regularity, Contributions to general algebra, 197-202 (1988) · Zbl 0699.20050
[16] Mcalister, D. B.: Regular Rees matrix semigroups and regular dubreil--jacotin semigroups, J. aust. math. Soc. (Series A) 31, 325-336 (1981) · Zbl 0474.06015
[17] Mcalister, D. B.: Rees matrix covers for locally inverse semigroups, Trans amer. Math. soc. 277, 727-738 (1983) · Zbl 0516.20039 · doi:10.2307/1999233
[18] Mcalister, D. B.: Rees matrix covers for regular semigroups, J. algebra 89, 264-279 (1984) · Zbl 0543.20041 · doi:10.1016/0021-8693(84)90217-5
[19] Mcalister, D. B.: Rees matrix covers for regular semigroups, Proceedings of 1984 marquette conference on semigroups, 131-141 (1985)
[20] Mcalister, D. B.: Quasi-ideal embeddings and Rees matrix covers for regular semigroups, J. algebra 152, 166-183 (1992) · Zbl 0770.20026 · doi:10.1016/0021-8693(92)90094-3
[21] Mitchell, B.: Theory of categories, (1965) · Zbl 0136.00604
[22] Neklyudova, V. V.: Polygons under semigroups with a system of local units, Fundam. appl. Math. 3, 879-902 (1997) · Zbl 0932.20056
[23] Neklyudova, V. V.: Morita equivalence of semigroups with a system of local units, Fundam. appl. Math. 5, 539-555 (1999) · Zbl 0963.20035 · http://mech.math.msu.su/~fpm/eng/99/992/99211h.htm
[24] B. Pécsi, On Morita contexts in bicategories, Preprint, www.renyi.hu/ aladar, 2009.
[25] Rees, D.: On semi-groups, Proc. camb. Philos. soc. 36, 387-400 (1940) · Zbl 0028.00401
[26] B. Steinberg, Strong Morita equivalence of inverse semigroups, Houston J. Math. (in press). · Zbl 1236.46049
[27] Talwar, S.: Morita equivalence for semigroups, J. aust. math. Soc. (Series A) 59, 81-111 (1995) · Zbl 0840.20067
[28] Talwar, S.: Strong Morita equivalence and a generalisation of the Rees theorem, J. algebra 181, 371-394 (1996) · Zbl 0855.20054 · doi:10.1006/jabr.1996.0125
[29] Talwar, S.: Strong Morita equivalence and the synthesis theorem, Internat. J. Algebra comput. 6, 123-141 (1996) · Zbl 0855.20055 · doi:10.1142/S0218196796000064