The structure of completely regular semigroups

Petrich, Mario

doi:10.1090/S0002-9947-1974-0330331-4

The structure of completely regular semigroups
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by Mario Petrich PDF

Trans. Amer. Math. Soc. 189 (1974), 211-236 Request permission

Abstract:

The principal result is a construction of completely regular semigroups in terms of semilattices of Rees matrix semigroups and their translational hulls. The main body of the paper is occupied by considerations of various special cases based on the behavior of either Green’s relations or idempotents. The influence of these special cases on the construction in question is studied in considerable detail. The restrictions imposed on Green’s relations consist of the requirement that some of them be congruences, whereas the restrictions on idempotents include various covering conditions or the requirement that they form a subsemigroup.

References

A. H. Clifford, Semigroups admitting relative inverses, Ann. of Math. (2) 42 (1941), 1037–1049. MR 5744, DOI 10.2307/1968781
A. H. Clifford, Bands of semigroups, Proc. Amer. Math. Soc. 5 (1954), 499–504. MR 62119, DOI 10.1090/S0002-9939-1954-0062119-9
A. H. Clifford, The structure of orthodox unions of groups, Semigroup Forum 3 (1971/72), no. 4, 283–337. MR 301117, DOI 10.1007/BF02572970
A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
P. H. H. Fantham, On the classification of a certain type of semigroup, Proc. London Math. Soc. (3) 10 (1960), 409–427. MR 121411, DOI 10.1112/plms/s3-10.1.409
J. M. Howie and G. Lallement, Certain fundamental congruences on a regular semigroup, Proc. Glasgow Math. Assoc. 7 (1966), 145–159. MR 197598, DOI 10.1017/S2040618500035334
Ján Ivan, On the decomposition of simple semigroups into a direct product, Mat.-Fyz. Časopis. Sloven. Akad. Vied. 4 (1954), 181–202 (Slovak, with Russian summary). MR 72881
Naoki Kimura, The structure of idempotent semigroups. I, Pacific J. Math. 8 (1958), 257–275. MR 102563, DOI 10.2140/pjm.1958.8.257
Gérard Lallement, Demi-groupes réguliers, Ann. Mat. Pura Appl. (4) 77 (1967), 47–129 (French). MR 225915, DOI 10.1007/BF02416940

The structure of bands of groups

Mario Petrich, Semigroups certain of whose subsemigroups have identities, Czechoslovak Math. J. 16(91) (1966), 186–198 (English, with Russian summary). MR 200370, DOI 10.21136/CMJ.1966.100723
Mario Petrich, The translational hull of a completely $0$simple semigroup, Glasgow Math. J. 9 (1968), 1–11. MR 229739, DOI 10.1017/S0017089500000239
Mario Petrich, A construction and a classification of bands, Math. Nachr. 48 (1971), 263–274. MR 294542, DOI 10.1002/mana.19710480119
Mario Petrich, Regular semigroups satisfying certain conditions on idempotents and ideals, Trans. Amer. Math. Soc. 170 (1972), 245–267. MR 304522, DOI 10.1090/S0002-9947-1972-0304522-0
Mario Petrich, Regular semigroups which are subdirect products of a band and a semilattice of groups, Glasgow Math. J. 14 (1973), 27–49. MR 313430, DOI 10.1017/S0017089500001701
Miyuki Yamada, Strictly inversive semigroups, Bull. Shimane Univ. Natur. Sci. 13 (1963), 128–138 (1964). MR 0393307
Miyuki Yamada, Inversive semigroups. III, Proc. Japan Acad. 41 (1965), 221–224. MR 186747
Miyuki Yamada and Naoki Kimura, Note on idempotent semigroups. II, Proc. Japan Acad. 34 (1958), 110–112. MR 98141