zbMATH — the first resource for mathematics

Embedding of regular semigroups in wreath products. (English) Zbl 0572.20045
A great deal of the author’s previous papers have addressed the problem of describing the structure of various types of regular semigroups. The present paper is a major step towards a more uniform and simplified treatment of this question. The main tool used in proving these structure theorems is that of embedding (regular) semigroups in wreath products of ”simpler” semigroups. The classes of regular semigroups for which structure descriptions are obtained in this way include the following: \({\mathcal H}\)-compatible inverse; inverse; \({\mathcal R}\)-unipotent; natural \({\mathcal R}\)-unipotent; generalized \({\mathcal L}\)-unipotent; left natural; \(\omega^ n\)-bisimple; \(\omega^ nI\)-bisimple, simple I-regular; \(\omega\) Y-inverse; \(\omega\) Y-\({\mathcal R}\)-unipotent; \(\omega\)-\({\mathcal R}\)- unipotent; generalized \(\omega\)-\({\mathcal L}\)-unipotent bisimple; orthodox; bisimple inverse; standard regular; and a few others.
This paper demonstrates the power of wreath product embeddings as a tool. It indicates a major step in understanding the structure of regular semigroups. On the other hand, its results do not provide a general uniform structure theory for regular semigroups yet. It seems to be one of the natural questions to ask for future research, how Nambooripad’s structure theory of regular semigroups translates into this framework.
Reviewer: H.Jürgensen

20M10 General structure theory for semigroups
Full Text: DOI
[1] Clifford, A.H.; Preston, G.B., The algebraic theory of semigroups, () · Zbl 0111.03403
[2] Eilenberg, S., Automata, languages, and machines, Vol. B, (1976), Academic Press New York
[3] Howie, J.M., An introduction to semigroup theory, (1976), Academic Press London · Zbl 0355.20056
[4] Krohn, K.; Rhodes, J.; Tilson, B., (), Chs. 5-9
[5] Rhodes, J., A homomorphism theorem for finite semigroups, Mathematical systems theory, 1, 289-304, (1967) · Zbl 0204.03303
[6] Rhodes, J., Infinite iteration of matrix semigroups, part II, (), write in care of J. Rhodes
[7] Warne, R.J., Matrix representation of d-simple semigroups, Trans. amer. math. soc., 106, 427-435, (1963) · Zbl 0115.02401
[8] Warne, R.J., A class of bisimple inverse semigroups, Pacific J. math., 18, 563-577, (1966) · Zbl 0161.01802
[9] Warne, R.J., The idempotent separating congruences of a bisimple inverse semigroup with identity, Publ. math. debrecen, 13, 203-206, (1966) · Zbl 0166.27801
[10] Warne, R.J., Bisimple inverse semigroups mod groups, Duke math. J., 34, 787-811, (1967) · Zbl 0168.26903
[11] Warne, R.J., I-bisimple semigroups, Trans. amer. math. soc., 130, 367-386, (1968) · Zbl 0153.34604
[12] Warne, R.J., Congruences on ω^n-bisimple semigroups, J. austral. math. soc., 257-274, (1969) · Zbl 0185.04403
[13] Warne, R.J., Ω^nI-bisimple semigroups, Acta math. acad. sci. hung., 21, 121-150, (1970) · Zbl 0198.04002
[14] Warne, R.J., I-regular semigroups, Math. japonicae, 15, 91-100, (1970) · Zbl 0248.20074
[15] Warne, R.J., Some properties of simple I-regular semigroups, Composito math., 22, 181-195, (1970) · Zbl 0205.02302
[16] Warne, R.J., y-unipotent semigroups, Nigerian J. science, 5, 245-248, (1972) · Zbl 0366.20050
[17] Warne, R.J., Generalized y-unipotent semigroups, Bull. Della unione math. ital., 5, 43-47, (1972) · Zbl 0283.20040
[18] Warne, R.J., Bands of maximal left groups, Rev. roumaine math. pures appl., 27, 1705-1707, (1972) · Zbl 0267.20062
[19] Warne, R.J., ΩY-y-unipotent semigroups, Jn̄ābha, 3, 99-118, (1973) · Zbl 0298.20048
[20] Warne, R.J., Generalized ω-y-unipotent bisimple semigroups, Pacific J. math., 51, 631-648, (1974) · Zbl 0306.20073
[21] Warne, R.J., Standard regular semigroups, Pacific J. math., 65, 539-562, (1976) · Zbl 0317.20033
[22] Warne, R.J., Natural regular semigroups, Colloq. math. soc.János bolyai, 20, 685-720, (1976), (Szeged) · Zbl 0317.20033
[23] Warne, R.J., On the structure of standard regular semigroups, Acta sci. math. szged, 41, 435-443, (1979) · Zbl 0394.20045
[24] Warne, R.J., On the structure of certain classes of regular semigroups, Math. japonicae, 24, 579-600, (1980) · Zbl 0444.20052
[25] Tilson, B., On the complexity of finite semigroups, J. pure appl. algebra, 5, 187-208, (1974) · Zbl 0293.20049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.