On left $P$-Ehresmann and right Ehresmann semigroups: $\lambda$-semidirect products and Zappa-Szép products

Year 2024, Volume: 53 Issue: 4, 1039 - 1059, 27.08.2024

Baddi Ul Zaman

https://doi.org/10.15672/hujms.1189391

Abstract

We show that the $\lambda$-semidirect product $M \rtimes^{\lambda} W$ of a left $P$-Ehresmann semigroup $M$ and a left restriction semigroup $W$ is a left $P$-Ehresmann semigroup. We explore the behavior of generalized Green's relations on $M \rtimes^{\lambda} W$, and investigate some properties of $M \rtimes^{\lambda} W$. Then the Zappa-Szép product of a right Ehresmann semigroup and its distinguished semilattice is studied. An example of Zappa-Szép product in the context of right Ehresmann semigroups is also given.

Keywords

generalized Green’s relations , left P-Ehresmann semigroup , right Ehresmann semigroup , Zappa-Szép product

References

• [1] A. Batbedat, $\gamma$-demi-groupes, demi-modules, produit demi-direct, in: Semigroups Proceedings, Lecture Notes in Mathematics 855, 1-18, Springer, Oberwolfach, 1981.
• [2] A. Batbedat and J.B. Fountain, Connections between left adequate semigroups and $\gamma$-semigroups, Semigroup Forum 22 (1), 59-65, 1981.
• [3] B. Billhardt, Extensions of semilattices by left type-A semigroups, Glasgow Math. J. 39 (1), 7-16, 1997.
• [4] B. Billhardt, On a wreath product embedding and idempotent pure congruences on inverse semigroups, Semigroup Forum 45 (1), 45-54, 1992.
• [5] M.G. Brin, On the Zappa-Szép product, Commun. Algebra 33 (2), 393-424, 2005.
• [6] J.R.B. Cockett and S. Lack, Restriction categories I: categories of partial maps, Theor. Comput. Sci. 270 (1-2), 223-259, 2002.
• [7] C. Cornock and V. Gould, Proper two-sided restriction semigroups and partial actions, J. Pure Appl. Algebra 216 (4), 935-949, 2012.
• [8] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 2002.
• [9] M. Erné and V. Joshi, Ideals in atomic posets, Discrete Math. 338 (6), 954-971, 2015.
• [10] J.B. Fountain, G.M.S. Gomes and V. Gould, The free ample monoid, Int. J. Algebra Comput. 19 (4), 527-554, 2009.
• [11] G.M.S. Gomes, Proper extensions of weakly left ample semigroups, Acta Math. Hung. 109 (1-2), 33-51, 2005.
• [12] G.M.S. Gomes and V. Gould, Graph expansions of unipotent monoids, Commun. Algebra 28 (1), 447-463, 2000.
• [13] G.M.S. Gomes and V. Gould, Proper weakly left ample semigroups, Int. J. Algebra Comput. 9 (6), 721-739, 1999.
• [14] V. Gould, Notes on restriction semigroups and related structures; formerly (weakly) left E-ample semigroups, 2010. See http://www-users.york.ac.uk/ ~varg1/restriction.pdf
• [15] V. Gould, Restriction and Ehresmann semigroups, in: Proceedings of the International Conference on Algebra 2010, 265-288, World Sci. Publ., Hackensack, 2012.
• [16] V. Gould and R. Zenab, Restriction semigroups and $\lambda$-Zappa-Szép products, Period. Math. Hung. 73 (2), 179-207, 2016.
• [17] C. Hollings, From right PP monoids to restriction semigroups: a survey, Eur. J. Pure Appl. Math. 2 (1), 21-57, 2009.
• [18] J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, 1995.
• [19] M. Jackson and T. Stokes, An invitation to C-semigroups, Semigroup Forum 62 (2), 279-310, 2001.
• [20] P.R. Jones, A common framework for restriction semigroups and regular - semigroups, J. Pure Appl. Algebra 216 (3), 618-632, 2012.
• [21] M. Kunze, Zappa products, Acta Math. Hung. 41 (3-4), 225-239, 1983.
• [22] M.V. Lawson, On Ehresmann semigroups, Semigroup Forum 103 (3), 953-965, 2021.
• [23] M.V. Lawson, Semigroups and ordered categories. I: The reduced case, J. Algebra 141 (2), 422-462, 1991.
• [24] E. Manes, Guarded and banded semigroups, Semigroup Forum 72 (1), 94-120, 2006.
• [25] J. Szép, On factorisable, not simple groups, Acta Univ. Szeged 13, 239-241, 1950.
• [26] V.S. Trokhimenko, Menger’s function systems, Izv. Vysš. Ucebn. Zaved. Mat. 11 (138), 71-78, 1973. (Russian)
• [27] G. Zappa, Sulla construzione dei gruppi prodotto di due sottogruppi permutabili tra loro, Atti Secondo Congresso Un. Ital. Bologna, 119-125, 1940.
• [28] R. Zenab, Algebraic properties of Zappa-Szép products of semigroups and monoids, Semigroup Forum 96 (2), 316-332, 2018.
• [29] R. Zenab, Decomposition of semigroups into semidirect and Zappa-Szép products, PhD thesis, University of York, 2014.