zbMATH — the first resource for mathematics

Proper restriction semigroups – semidirect products and \(W\)-products. (English) Zbl 1295.20064
Left restriction semigroups are unary semigroups isomorphic to unary subsemigroups of partial transformation semigroups, where the unary operation is \(x\to x^+=I_{\mathrm{dom}\;x}\). The semilattice of projections of a left restriction semigroup \(S\) is \(E=\{x^+,\;x\in S\}\); \(\sigma_S\) is the least congruence identifying all the elements of \(E\). A left restriction semigroup is proper if \(a^+=b^+\), \(a\sigma_Sb\) yields \(a=b\), and left ample if \(xz=yz\) yields \(xz^+=yz^+\).
It is known that any proper left ample semigroup embeds into a so-called \(W\)-product, which is a subsemigroup of a reverse semidirect product of a semilattice \(\mathcal Y\) by a monoid \(T\), where the action of \(T\) on \(\mathcal Y\) is injective with images of the action being order ideals of \(\mathcal Y\). Here necessary and sufficient conditions are given for a proper left restriction semigroup to be embeddable into a \(W\)-product.

20M10 General structure theory for semigroups
Full Text: DOI
[1] Branco, M. 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] Cornock, C.; Gould, V., Proper restriction semigroups and partial actions, J. Pure Appl. Algebra, 216, 935-949, (2012) · Zbl 1258.20047
[3] Fountain, J., A class of right PP monoids, Quart. J. Math. Oxford, 28, 285-300, (1977) · Zbl 0377.20051
[4] Fountain, J.; Gomes, G. M. S., Proper left type-A monoids revisited, Glasgow Math. J., 35, 293-306, (1993) · Zbl 0802.20051
[5] Fountain, J.; Gomes, G. M. S.; Gould, V., The free ample monoid, Int. J. Algebra Comp., 19, 527-554, (2009) · Zbl 1192.20041
[6] Gomes, G. M. S.; Gould, V., Proper weakly left ample semigroups, Int. J. Algebra Comp., 9, 721-739, (1999) · Zbl 0948.20036
[7] Gomes, G. M. S.; Szendrei, M. B., Almost factorizable weakly ample semigroups, Comm. Algebra, 35, 3503-3523, (2007) · Zbl 1144.20037
[8] V. Gould, Notes on restriction semigroups and related structures, http://www-users.york.ac.uk/ varg1/restriction.pdf.
[9] P. A. Grillet, Commutative Semigroups, Kluwer Academic Publishers (2001).
[10] Hollings, C. D., From right PP monoids to restriction semigroups: a survey, European J. Pure Appl. Math., 2, 21-57, (2009) · Zbl 1214.20056
[11] J. M. Howie, Fundamentals of Semigroup Theory, Oxford Science Publications (1995). · Zbl 0835.20077
[12] Lawson, M. V., The structure of type A semigroups, Quart. J. Math. Oxford, 37, 279-298, (1986) · Zbl 0605.20057
[13] Munn, W. D., A note on E-unitary inverse semigroups, Bull. London Math. Soc., 8, 71-76, (1976) · Zbl 0347.20036
[14] M. B. Szendrei, Embedding into almost left factorizable restriction semigroups, Comm. Algebra, to appear. · Zbl 1275.20064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.