Abstract
The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12, 13]. Of this, we give a new proof.
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Anderson, O.,Ein Bericht über die Struktur abstrakter Halbgruppen, Thesis (Staatsexamenserbeit), Hamburg, 1952.
Chrislock, J.,On medial semigroups, J. Algebra 12 (1969), 1–9.
Clifford, A.H.,Semigroups admitting relative inverses, Annals of Math. 42 (1941), 1037–1049.
Clifford, A.H. and G.B. Preston,The algebraic theory of semigroups, vol. 1, Amer. Math. Soc. Providence, Rhode Island, 1961.
Croisot, R.,Demi-groupes inversifs et demi-groupes réunions de demi-groupes simples. Ann. Sci. École Norm. Sup. (3) 70 (1953), 361–379.
Ivan, J.,On the decomposition of simple semigroups into a direct product, Mat.-Fyz. Casopis Slovensk. Akad. Vied 4 (1954), 181–202.
Iyengar, H.R. Krishna,Semilattice of bisimple regular semigroups, Proc. Amer. Math. Soc. 28 (1971), 361–365.
Munn, W. D.,Pseudo inverses in semigroups, Proc. Cambridge Phil. Soc. 57 (1961), 247–250.
Petrich, M.,The maximal semilattice decomposition of a semigroup, Math. Z. 85 (1964), 68–82.
Putcha, M.S. and J. Weissglass,A semilattice decomposition into semigroups with at most one idempotent, Pacific. J. Math. 39 (1971), 225–228.
Schein, B.M.,Homomorphisms and subdirect decompositions of semigroups, Pacific J. Math. 17 (1966), 529–547.
Tamura, T.,The theory of construction of finite semigroups I, Osaka Math. J. 8 (1956), 243–261.
Tamura, T.,Another proof of a theorem concerning the greatest semilattice decomposition of a semigroup. Proc. Japan Acad. 40 (1964), 777–780.
Tamura, T. and N. Kimura,On decomposition of a commutative semigroup, Kodai Math. Sem. Rep. 4 (1954), 109–112.
Tamura, T. and J. Shafer,On exponential semigroups I, Proc. Japan Acad. 48 (1972), 77–80.
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The results of this paper were obtained by the author between January and July of 1971, while an undergraduate at the University of California, Santa Barbara.
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Putcha, M.S. Semilattice decompositions of semigroups. Semigroup Forum 6, 12–34 (1973). https://doi.org/10.1007/BF02389104
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DOI: https://doi.org/10.1007/BF02389104