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Proper two-sided restriction semigroups and partial actions. (English) Zbl 1258.20047
An algebra $$(S,\cdot,+,*)$$ of type $$(2,1,1)$$ is called a (two-sided) restriction semigroup if $$(S,\cdot )$$ is a semigroup, $$(S,\cdot,+)$$ satisfies the identities $$x^+x=x$$, $$x^+y^+=y^+x^+$$, $$(x^+y^+)^+=x^+y^+$$, $$xy^+=(xy)^+x$$, $$(S,\cdot,*)$$ satisfies the dual identities, and $$(x^+)^*=x^+$$, $$(x^*)^+=x^*$$.
The structure of restriction semigroups is studied. Using both sides actions of a monoid on a semilattice, a proper restriction semigroup is constructed. To obtain the full class of proper restriction semigroups, partial actions of monoids are used. Thus the main result is a structure theorem for proper restriction semigroups.
Reviewer’s comment: The topic of the paper under review is close to that of a paper by M. B. Szendrei, [Int. J. Algebra Comput. 22, No. 3, 1250024 (2012; Zbl 1258.20048)].

##### MSC:
 20M10 General structure theory for semigroups 20M30 Representation of semigroups; actions of semigroups on sets
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##### References:
 [1] Branco, M.J.J.; Gomes, G.M.S.; Gould, V., Extensions and covers for semigroups whose idempotents form a left regular band, Semigroup forum, 81, 51-70, (2010) · Zbl 1237.20055 [2] Cockett, J.R.; Guo, X., Stable meet semilattice fibrations and free restriction categories, Theor. appl. comput. sci., 16, 307-341, (2006) · Zbl 1106.18003 [3] Cockett, J.R.; Lack, S., Restriction categories I: categories of partial maps, Theoret. comput. sci., 270, 223-259, (2002) · Zbl 0988.18003 [4] C. Cornock, Restriction semigroups: structure, varieties and presentations, Ph.D. Thesis, University of York, 2011. [5] Fountain, J., Right PP monoids with central idempotents, Semigroup forum, 13, 229-237, (1977) · Zbl 0353.20051 [6] Fountain, J., A class of right PP monoids, Quart. J. math. Oxford, 28, 285-300, (1977) · Zbl 0377.20051 [7] Fountain, J.; Gomes, G.M.S., Proper covers of ample monoids, Proc. edinb. math. soc., 49, 277-289, (2006) · Zbl 1107.20050 [8] Fountain, J.; Gomes, G.M.S.; Gould, V., The free ample monoid, Internat. J. algebra comput., 19, 527-554, (2009) · Zbl 1192.20041 [9] Gomes, G.M.S.; Gould, V., Proper weakly left ample semigroups, Internat. J. algebra comput., 9, 721-739, (1999) · Zbl 0948.20036 [10] Gomes, G.M.S.; Gould, V., Graph expansions of unipotent monoids, Comm. algebra, 28, 447-473, (2000) · Zbl 0965.20034 [11] V. Gould, Notes on restriction semigroups and related structures, preprint at, http://www-users.york.ac.uk/ varg1/restriction.pdf. [12] Hollings, C.D., From right PP monoids to restriction semigroups: a survey, Eur. J. pure appl. math., 2, 1, 21-57, (2009) · Zbl 1214.20056 [13] Howie, J.M., Fundamentals of semigroup theory, (1995), Oxford Science Publications · Zbl 0835.20077 [14] Jackson, M.; Stokes, T., An invitation to $$C$$-semigroups, Semigroup forum, 62, 279-310, (2001) · Zbl 0982.20051 [15] P.R. Jones, A common framework for restriction semigroups and regular $$\ast$$-semigroups, J. Pure Appl. Algebra, in press (doi:10.1016/j.jpaa.2011.07.014). · Zbl 1257.20058 [16] Lawson, M.V., The structure of type A semigroups, Quart. J. math. Oxford, 37, 2, 279-298, (1986) · Zbl 0605.20057 [17] McAlister, D.B., Groups, semilattices and inverse semigroups, Trans. amer. math. soc., 192, 227-244, (1974) · Zbl 0297.20071 [18] McAlister, D.B., Groups, semilattices and inverse semigroups II, Trans. amer. math. soc., 196, 351-370, (1974) · Zbl 0297.20072 [19] Manes, E., Guarded and banded semigroups, Semigroup forum, 72, 94-120, (2006) · Zbl 1097.20044 [20] Munn, W.D., A note on E-unitary inverse semigroups, Bull. lond. math. soc., 8, 71-76, (1976) · Zbl 0347.20036 [21] O’Carroll, L., Embedding theorems for proper inverse semigroups, J. algebra, 42, 26-40, (1976) · Zbl 0343.20037 [22] Petrich, M.; Reilly, N.R., A representation of E-unitary inverse semigroups, Quart. J. math. Oxford, 30, 339-350, (1979) · Zbl 0412.20058 [23] Schein, B.M., Restrictively multiplicative algebras of transformations, Izv. vysš. Učebn. zaved. matematika, 4, 95, 91-102, (1970), (in Russian) [24] Schein, B.M., Relation algebras and function semigroups, Semigroup forum, 1, 1-62, (1970) · Zbl 0197.29404
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