The structure of completely regular semigroups

Petrich, Mario

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

The structure of completely regular semigroups

Author: Mario Petrich
Journal: Trans. Amer. Math. Soc. 189 (1974), 211-236
MSC: Primary 20M10
DOI: https://doi.org/10.1090/S0002-9947-1974-0330331-4
MathSciNet review: 0330331
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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.

[1] A. H. Clifford, Semigroups admitting relative inverses, Ann. of Math. (2) 42 (1941), 1037-1049. MR 3, 199. MR 0005744 (3:199c)
[2] -, Bands of semigroups, Proc. Amer. Math. Soc. 5 (1954), 499-504. MR 15, 930. MR 0062119 (15:930c)
[3] -, The structure of orthodox unions of groups, Semigroup Forum 3 (1972), 283-337. MR 0301117 (46:275)
[4] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Math. Surveys, no.7, Amer. Math. Soc., Providence, R.I., 1961. MR 24 #A2627. MR 0132791 (24:A2627)
[5] P. H. H. Fantham, On the classification of a certain type of semigroup, Proc. London Math. Soc. (3) 10 (1960), 409-427. MR 22 #12149. MR 0121411 (22:12149)
[6] J. M. Howie and G. Lallement, Certain fundamental congruences on a regular semigroup, Proc. Glasgow Math. Assoc. 7 (1966), 145-159. MR 33 #5763. MR 0197598 (33:5763)
[7] J. Ivan, On the decomposition of simple semigroups into a direct product, Mat.-Fyz. Časopis 4 (1954), 181-202. (Slovak) MR 17, 346. MR 0072881 (17:346c)
[8] N. Kimura, The structure of idempotent semigroups. I, Pacific J. Math. 8 (1958), 257-275. MR 21 #1354. MR 0102563 (21:1354)
[9] G. Lallement, Demi-groupes réguliers, Ann. Mat. Pura Appl. (4) 77 (1967), 47-129. MR 37 #1505. MR 0225915 (37:1505)
[10] J. Leech, The structure of bands of groups (to appear).
[11] M. Petrich, Semigroups certain of whose subsemigroups have identities, Czechoslovak J. Math. 16 (91) (1966), 186-198. MR 34 #266. MR 0200370 (34:266)
[12] -, The translational hull of a completely 0-simple semigroup, Glasgow Math. J. 9 (1968), 1-11. MR 37 #5313. MR 0229739 (37:5313)
[13] -, A construction and a classification of bands, Math. Nachr. 48 (1971), 263-274. MR 0294542 (45:3612)
[14] -, Regular semigroups satisfying certain conditions on idempotents and ideals, Trans. Amer. Math. Soc. 170 (1972), 245-267. MR 0304522 (46:3657)
[15] -, Regular semigroups which are subdirect products of a band and a semilattice of groups, Glasgow Math. J. 14 (1973), 27-49. MR 0313430 (47:1984)
[16] M. Yamada, Strictly inversive semigroups, Bull. Shimane Univ. 13 (1964), 128-138. MR 0393307 (52:14117)
[17] -, Inversive semigroups. III, Proc. Japan Acad. 41 (1965), 221-224. MR 32 #4204. MR 0186747 (32:4204)
[18] M. Yamada and N. Kimura, Note on idempotent semigroups. II, Proc. Japan Acad. 34 (1958), 110-112. MR 20 #4603. MR 0098141 (20:4603)