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 for factorisable semigroups. (English) Zbl 0991.20046

Let S be a semigroup with set of idempotents E=E(S). Then S is called factorisable if S=S 2 ; S is called sandwich if S=SES; S is called local units if S=ESE.

S-Act denotes the category of left S-acts, US-Act={ S MS-ActSM=M}. US-FAct={MUS-Actζ m =ε}, where ζ M ={(m 1 ,m 2 )sm 1 =sm 2 ,sS} is an S-congruence on S M, and ζ S a two-sided congruence on S.

The authors prove the following theorems: Let R,S be factorisable semigroups. LetUR-FAct G F US-FAct be a category inverse equivalence, R M=G( S S ' ) and S N=F( R R ' ). Then R M S and S N R are unitary biacts such that (i) R M and S N are generators in UR-FAct and US-FAct respectively; (ii) FSHom R ( R M,-) and GRHom S ( S N,-), where the functors are defined in the obvious way and “” means the natural isomorphism; (iii) S ' SHom R ( R M, R M) and R ' RHom S ( S N, S N) as semigroups; (iv) R M S RHom S ( S N R , S S ' ) and S N R SHom R ( R M S , R R ' ) as biacts; (v) if we identify S N R with SHom R ( R M S , R R ' ), identify S ' with a right ideal of Hom R ( R M, R M)Hom S ( S S ' , S S ' ) and define

:M S ' NR ' ,mnm,n=(m)nand:N R ' MS ' ,nmn,m,

where n,m: R M R M, x(x)n·m, then (R ' ,S ' , R ' M S ' , S ' M R ' ,,) is a unitary Morita context with and surjective. Let R,S be factorisable semigroups. If there exists a unitary Morita context (R,S, R P S , S Q R ,,) with and surjective, then we have the following category inverse equivalence UR-FAct G F US-FAct, where F=Q R -/ζ Q R - and G=P S -/ζ P S - .

Let R,S be factorisable semigroups. Then the category UR-FAct is equivalent to the category US-FAct if and only if there exists a unitary Morita context (R ' ,S ' , R ' P S ' , S ' Q R ' ,,) with and surjective. Moreover, if this is the case, then we have the following category inverse equivalence: UR-FAct G F US-FAct, where F=SHom R ( R P,-) and G=RHom S ( S Q,-). If there exists a unitary Morita context (R,S, R P S , S Q R ,,) with and surjective, then R and S are said to be strongly Morita equivalent.

Define the category S-FxAct={MUS-ActΓ M ' is an S-isomorphism} where Γ M ' :SSHom S (S,M)M, st·φ(st)φ.

Now the authors prove: (i) If either iI SS-FxAct for any index set I or SUS-FAct holds, then S is strongly Morita equivalent to a monoid if and only if S=SeS for some eE. Moreover, if this is the case, then S is strongly Morita equivalent to eSe. (ii) If iI SS-FxAct for any index set I, then the following statements are equivalent: (a) S=SES; (b) S is strongly Morita equivalent to a sandwich semigroup; (c) S is strongly Morita equivalent to a local units semigroup. Let S be an arbitrary semigroup. Then S is a completely simple semigroup if and only if iI SS-FxAct for any index set I and S is strongly Morita equivalent to a group.

Let S be a like unity semigroup. Then the following statements hold: (i) S is strongly Morita equivalent to a group if and only if S is a completely simple semigroup; (ii) S is strongly Morita equivalent to a monoid if and only if S=SeS for some eE; (iii) S is strongly Morita equivalent to a local semigroup if and only if S=SES. Here a semigroup is called like unity if xSxxS for any xS.

MSC:
20M50Connections of semigroups with homological algebra and category theory
20M30Representation of semigroups; actions of semigroups on sets
References:
[1]G. D. Abrams, Morita equivalennce for rings with local units, Comm Algebra, 1983, 11:801–837 · Zbl 0503.16034 · doi:10.1080/00927878308822881
[2]P. N. Anh, L. Marki, Morita equivalennce for rings without identity, Tsukuba J. Math., 1987, 11:1–16
[3]H. Bass, The Morita theorem, lecture notes, University of Oregan, 1962
[4]J. L. Garcia, J. J. Simon, Morita equivalence for idempotent rings, J. Pure. Appl. Algebra, 1991, 76:39–56 · Zbl 0747.16007 · doi:10.1016/0022-4049(91)90096-K
[5]K. Morita, Category-isomorphism and endomorphism rings of modules, Trans. Amer. Math. Soc., 1961, 103:451–469 · doi:10.1090/S0002-9947-1962-0140560-2
[6]Y. H. Xu, K. P. Shum, Morita equivalence for infinite matrix rings, Comm. Algebra, 1999, 27:1751–1782 · Zbl 0947.20012 · doi:10.1080/00927879908826755
[7]U. Knauer, Projectivity of acts and Morita equivalence of monoids, Semigroup Forum, 1972, 3:359–370 · Zbl 0231.18013 · doi:10.1007/BF02572973
[8]B. Banaschewski, Functors into the category of M-sets, Abh. Math. Sem. Univ. Hamberg, 1972, 8:49–64 · Zbl 0257.18011 · doi:10.1007/BF02996922
[9]U. Knauer, P. Normak, Morita duality for monoids, Semigroup Forum, 1990, 40:39–57 · Zbl 0689.20049 · doi:10.1007/BF02573250
[10]S. Talwar, Morita equivalence for semigroups, J. Austral Soc. (Series A), 1995, 59:81–111 · doi:10.1017/S1446788700038489
[11]S. Talwar, Strong Morita equivalence and a generalisation of the Rees theorem, J. Algebra, 1996, 181:371–394 · Zbl 0855.20054 · doi:10.1006/jabr.1996.0125
[12]J. M. Howie, An Introduction to Semigroup Theory, London: Academic Press, 1976
[13]F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Berlin: Springer, 1974
[14]J. J. Rotman, An Introduction to Homological Algebra, London: Academic Press Inc, 1979
[15]N. Jacobson, Basic Algebra II, San Francisco: Freeman, 1980
[16]Y. Q. Chen, K. P. Shum, Projective and indecomposable S-acts, Science in China (Series A), 1999, 42(6):593–599 · Zbl 0957.20049 · doi:10.1007/BF02880077