Abstract
In a result generalising the Ehresmann–Schein–Nambooripad Theorem relating inverse semigroups to inductive groupoids, Lawson has shown that Ehresmann semigroups correspond to certain types of ordered (small) categories he calls Ehresmann categories. An important special case of this is the correspondence between two-sided restriction semigroups and what Lawson calls inductive categories. Gould and Hollings obtained a one-sided version of this last result, by establishing a similar correspondence between left restriction semigroups and certain ordered partial algebras they call inductive constellations (a general constellation is a one-sided generalisation of a category). We put this one-sided correspondence into a rather broader setting, at its most general involving left congruence D-semigroups (which need not satisfy any semiadequacy condition) and what we call co-restriction constellations, a finitely axiomatized class of partial algebras. There are ordered and unordered versions of our results. Two special cases have particular interest. One is that the class of left Ehresmann semigroups (the natural one-sided versions of Lawson’s Ehresmann semigroups) corresponds to the class of co-restriction constellations satisfying a suitable semiadequacy condition. The other is that the class of ordered left Ehresmann semigroups (which generalise left restriction semigroups and for which semigroups of binary relations equipped with domain operation and the inclusion order are important examples) corresponds to a class of ordered constellations defined by a straightforward weakening of the inductive constellation axioms.
Similar content being viewed by others
References
Batbedat, A.: \(\gamma \)-demi-groups, demi-modules, produit demi-directs. In: Jürgensen, H., Petrich, M., Weinert, H.J. (eds.) Semigroups Proceedings, Oberwolfalch, Germany 1978, Lecture Notes in Mathematics 855, pp. 1–18. Springer-Verlag, Berlin, Heidelberg, New York (1981)
Berberian, S.K.: Baer rings and Baer \(*\)-rings. The University of Texas at Austin, Registered U.S. Copyright Office March 1988, revised edition (2003)
Branco, M.J.J., Gomes, G.M.S., Gould, V.: Left adequate and left Ehresmann monoids. Int. J. Algebra Comput. 21, 1259–1284 (2011)
Bredikhin, D.A.: Abstract Characterisation of Some Classes of Binary Relation Algebras. Algebra and Number Theory No. 2, pp. 3–19. Kabardino-Balkarsk. Gos. Univ., Nalchik (1977) (In Russian)
Fountain, J., Gomes, G.M.S., Gould, V.: A Munn type representation for a class of E-semiadequate semigroups. J. Algebra 218, 693–714 (1999)
Gomes, G.M.S., Gould, V.: Left adequate and left Ehresmann monoids II. J. Algebra 348, 171–196 (2011)
Gould, V., Hollings, C.: Restriction semigroups and inductive constellations. Commun. Algebra 38, 261–287 (2009)
Gould, V., Hollings, C.: Partial actions of inverse and weakly left E-ample semigroups. J. Aust. Math. Soc. 86, 355–377 (2009)
Gould, V., Stokes, T.: Constellations and their relationship to categories. To appear in Algebra Universalis
Grätzer, G.: Universal Algebra, 2nd edn. Springer, New York (1979)
Jackson, M., Stokes, T.: An invitation to C-semigroups. Semigroup Forum 62, 279–310 (2001)
Jackson, M., Stokes, T.: Agreeable semigroups. J. Algebra 266, 393–417 (2003)
Kambites, M.: Retracts of trees and free left adequate semigroups. Proc. Edinb. Math. Soc. 54, 731–747 (2011)
Lawson, M.V.: Semigroups and ordered categories I: the reduced case. J. Algebra 141, 422–462 (1991)
Maeda, S.: On the lattice of projections of a Baer \(*\)-ring. J. Sci. Hiroshima Univ. Ser. A 22, 75–88 (1958)
Manes, E.: Guarded and banded semigroups. Semigroup Forum 72, 94–120 (2006)
Möller, B., Struth, G.: Algebras of modal operators and partial correctness. Theor. Comput. Sci. 351, 221–239 (2006)
Resende, P.: Étale groupoids and their quantales. Adv. Math. 208, 147–209 (2007)
Stokes, T.: Domain and range operations in semigroups and rings. Commun. Algebra 43, 3979–4007 (2015)
Trokhimenko, V.S.: Mengers function systems. Izv. Vysš. Učebn. Zaved. Matematika 11, 71–78 (1973). (In Russian)
Acknowledgements
The author gratefully acknowledges Professor Victoria Gould for the many helpful discussions whilst he was on sabbatical at the University of York in mid-2014, and further interactions since. The current work may be viewed as part of a larger collaboration with Professor Gould on the subject of constellations.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Victoria Gould.
Rights and permissions
About this article
Cite this article
Stokes, T. D-semigroups and constellations. Semigroup Forum 94, 442–462 (2017). https://doi.org/10.1007/s00233-017-9851-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-017-9851-z