Zappa-Szép Products of Semigroups


The internal Zappa-Szép products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szép products of semigroups. We consider the structure of the internal Zappa-Szép product as an enlargement. We show how rectangular band can be described as the Zappa-Szép product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szép product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szép product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szép product of a semilattice E and a group G by constructing an inductive groupoid.

Share and Cite:

Wazzan, S. (2015) Zappa-Szép Products of Semigroups. Applied Mathematics, 6, 1047-1068. doi: 10.4236/am.2015.66096.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Zappa, G. (1940) Sulla construzione dei gruppi prodotto di due dadi sottogruppi permutabili tra loro. Atti del secondo congresso dell’Unione Matematica Italiana, Cremonese, Roma, 119-125.
[2] Lawson, M. (2008) A Correspondence between a Class of Monoids and Self-Similar Group Actions I. Semigroup Forum, 76, 489-517.
[3] Brownlowe, N., Ramagge, J., Robertson, D. and Whittaker, M. (2014) Zappa-Szép Products of Semigroups and Their C*-Algebras. Journal of Functional Analysis, 266, 3937-3967.
[4] Gould, V. and Zenab, R. (2013) Semigroups with Inverse Skeletons and Zappa-Szép Products. CGASA, 1, 59-89.
[5] Gilbert, N.D. and Wazzan, S. (2008) Zappa-Szép Products of Bands and Groups. Semigroup Forum, 77, 438-455.
[6] Wazzan, S.A. (2008) The Zappa-Szép Product of Semigroups. PhD Thesis, Heriot-Watt University, Edinburgh.
[7] Lawson, M.V. (1998) Inverse Semigroups. World Scientific, Singapore City.
[8] Lawson, M.V. (1996) Enlargements of Regular Semigroups. Proceedings of the Edinburgh Mathematical Society (Series 2), 39, 425-460.
[9] Lawson, M.V. and Marki, L. (2000) Enlargement and Covering by Rees Matrix Semigroups. Springer-Verlag, Berlin, 191-195.
[10] Kunze, M. (1983) Zappa Products. Acta Mathematica Hungarica, 41, 225-239.
[11] Nico, W.R. (1983) On the Regularity of Semidirect Products. Journal of Algebra, 80, 29-36.
[12] Hall, T.E. (1982) Some Properties of local Subsemigroups Inherited by Larger Subsemigroups. Semigroup Forum, 25, 35-49.
[13] Fiedorowicz, Z. and Loday, J.L. (1991) Crossed Simplicial Groups and Their Associated Homology. Transactions of the American Mathematical Society, 326, 57-87.
[14] Kassel, C. (1995) Quantum Groups. Graduate Texts in Mathematics, No. 155, Springer, New York.
[15] Billhardt, B. (1992) On a Wreath Product Embedding and Idempotent Pure Congruences on Inverse Semigroups. Semigroup Forum, 45, 45-54.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.