zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 μ X :SXX given by μ X (sx)=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-𝐅𝐀𝐜𝐭. Two semigroups S and T are called ‘Morita equivalent’ if the categories S-𝐅𝐀𝐜𝐭 and T-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:Obj(C)×Obj(C)Mor(C), q(A,B):BA, 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 xy=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,-,-,[-,-]) with surjective mappings; 4) there is a consolidation q on C(S) and a local isomorphism C(S) q 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
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.
[6]Howie, J. M.: Fundamentals of semigroup theory, (1995)
[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)
[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)
[12]Lane, S. Mac: Categories for the working mathematician, (1998)
[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)
[23]Neklyudova, V. V.: Morita equivalence of semigroups with a system of local units, Fundam. appl. Math. 5, 539-555 (1999) · Zbl 0963.20035 · doi: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).
[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