Abstract
We introduce the notion of a fundamental semilattice of semigroups, which is a special case of a semilattice of semigroups and a generalization of a strong semilattice of semigroups. The relations among fundamental semilattices of semigroups, refined semilattices of semigroups (originally introduced by L. Zhang, K. P. Shum and R. H. Zhang [21]) and regular semilattices of semigroups (introduced by Z. P. Wang and Y. L. Zhou [17]) are discussed. As an application, we prove that a semigroup is completely regular if and only if it is a fundamental semilattice of completely simple semigroups. The relation between this structure theorem and the theorem on completely regular semigroups by Lallement is also illustrated. The structure of some special types of completely regular semigroups is also described as fundamental semilattices of completely simple semigroups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Bogdanović, Semigroups with a System of Subsemigroups, Institute of Mathematicsk (Novi Sad, Yugoslavia, 1985).
Clifford A.H.: Semigroups admitting relative inverses. Ann. Math., 42, 1037–1049 (1941)
Clifford A.H., Petrich M.: Some classes of completely regular semigroups. J. Algebra., 46, 462–480 (1977)
J. B. Fountain, Abundant semigroups, Proc. London Math. Soc., 44 (1982), 103–129.
P.-A. Grillet, Semigroups: An Introduction to the Structure Theory, Marcel Dekker, New York (1995).
J. M. Howie, An Introduction to Semigroup Theory, Academic Press (London, 1976).
X. Z. Kong and K. P. Shum, On the structure of regular crypto semigroups, Comm. Algebra, 29 (2001), 2461–2479.
G. Lallement, Demi-groupes réguliers, Ann. Mat. Pura Appl., 77 (1967), 47–129.
D. McLean, Idempotent semigroups, Amer. Math. Monthly, 61 (1954), 110–113.
M. Petrich, Regular semigroups satisfying certain conditions on idempotents and ideals, Trans. Amer. Math. Soc., 170 (1972), 245–267.
M. Petrich, Introduction to semigroups, Charles E. Merrill (Columbus, Ohio, 1973).
M. Petrich, The structure of completely regular semigroups, Trans. Amer. Math. Soc., 189 (1974), 211–236.
M. Petrich, A structure theorem for completely regular semigroups, Proc. Amer. Math. Soc., 99 (1987), 617–622.
M. Petrich and N. R. Reilly, Completely Regular Semigroups, John Wiley and Sons (New York, 1999).
Z. P. Wang, R. H. Zhang and M. Xie, Regular orthocryptou semigroups, Semigoup Forum, 69 (2004), 281–302.
Z. P.Wang, Y. Q. Guo and K. P. Shum, On refined semilattices of semigroups, Algebra Colloq., 15 (2008), 331–336.
Z. P. Wang and Y. L. Zhou, Regular semilattice of semigroups and its applications, Semigroup Forum, 87 (2013), 393–406.
M. Yamada and N. Kimura, Note on idempotent semigroups II, Proc. Japan Acad., 34 (1958), 110–112.
H. Y. Yu and Z. P. Wang, The refined semilattice construction of locally orthodox regular cryptogroups, Comm. Algebra, 40 (2012), 552–564.
H. Y. Yu and Z. P. Wang, Completely regular semigroups for which every congruence on each \({\mathcal{D}}\)-class extends to a congruence on the whole semigroup, Comm. Algebra, to appear.
L. Zhang, K. P. Shum and R. H. Zhang, Refined semilattices of semigroups, Algebra Colloq., 8 (2001), 93–108.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the National Natural Science Foundation of China (11101336) and the Fundamental Research Funds for the Central Universities (XDJK2013B012).
Rights and permissions
About this article
Cite this article
Wang, ZP. Fundamental semilattices of semigroups. Acta Math. Hungar. 146, 22–39 (2015). https://doi.org/10.1007/s10474-015-0504-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-015-0504-y
Key words and phrases
- semilattice of semigroups
- strong semilattice of semigroups
- fundamental semilattice of semigroups
- refined semilattice of semigroups
- regular semilattice of semigroups
- completely regular semigroup