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Infinite iteration of matrix semigroups. II: Structure theorem for arbitrary semigroups up to aperiodic morphism. (English) Zbl 0626.20050
This paper is Part II of a nearly monograph-sized study of the “global theory of semigroups” [Part I, ibid. 98, 422-451 (1986; Zbl 0584.20053)]. In part I, a structure theory for torsion semigroups was derived. It is based on an iterated matrix construction. In this second part, the structure theory for arbitrary semigroups is developed.
The “global theory of semigroups” will, given a semigroup T, determine semigroups S and X and a surjective morphism \(\vartheta\) such that X is “easily globally computed”, S is a “special” subsemigroup of X, and \(\vartheta\) is a “fine” morphism of S onto T. In this paper, “special” means “equal”, that is, \(S=X\); “fine” means “aperiodic”, that is, \(\vartheta\) is aperiodic if and only if the pre- images of aperiodic subsemigroups of T are aperiodic subsemigroups of S; finally and roughly speaking, a semigroup X is “easily globally computed” in the context of this paper, if it is a cyclic monoid, a Rees matrix semigroup over easily globally computed semigroups with countable index sets, or an elementary projective limit of easily globally computed semigroups - at least, this is the construction for the countable case; a modified definition is also given for the uncountable case. Thus, easily globally computed semigroups in the sense of this paper are infinite iterations of matrix semigroups over cyclic monoids.
The main result of this paper is as follows: For any given monoid T there exist a monoid X and a surjective morphism \(\vartheta\) : \(X\to T\) such that X is an infinite iteration of matrix semigroups over cyclic monoids and \(\vartheta\) is aperiodic. The result implies that the maximal subgroups of X are finite cyclic groups.
In the course of proving this and some related results, quite a few statements of independent importance are derived. Clearly this paper is an outstanding contribution to the structure theory of semigroups and, furthermore, to the discussion concerning its methodology. The reviewer looks forward to seeing Parts III and IV.
Reviewer: H.Jürgensen

MSC:
20M10 General structure theory for semigroups
20M20 Semigroups of transformations, relations, partitions, etc.
20M15 Mappings of semigroups
20M30 Representation of semigroups; actions of semigroups on sets
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[1] ()
[2] Birkhoff, G, On the structure of abstract algebras, (), 433-454 · JFM 61.1026.07
[3] Brzozowski, K.Culik; Gabrielian, A, Classification of noncounting events, J. comput. system sci., 5, 41-53, (1971) · Zbl 0241.94050
[4] Brzozowski, K, Open problems about regular languages, ()
[5] Cohn, P.M, Universal algebras, (1965), Harper & Row New York · Zbl 0141.01002
[6] Clifford, A.H; Preston, G.B; Clifford, A.H; Preston, G.B, The algebraic theory of semigroups, Amer. math. soc. mathematical surveys, Amer. math. soc. mathematical surveys, Vol. 7, (1967), Vols. 1 and 2 · Zbl 0178.01203
[7] Eilenberg, S, ()
[8] Lallement, G, ()
[9] Morse, M; Henlund, G.A, Unending chess, symbolic dynamics and a problem in semigroups, Duke math. J., 11, 1-7, (1944) · Zbl 0063.04115
[10] Rhodes, J, Infinite iteration of matrix semigroups. part I. structure theorem for torsion semigroups, J. alg., 98, 422-451, (1986) · Zbl 0584.20053
[11] \scJ. Rhodeset al., Prime Decomposition Theorem for Arbitrary Semigroups: General Holonomy Decomposition and Synthesis Theorem, in progress. · Zbl 0679.20056
[12] \scJ. Rhodes, Prime Decomposition Theorem for Arbitrary Semigroups: General Left Ideal and Subsemigroup Version, in preparation.
[13] \scJ. Rhodes and B. Tilson, The Kernel of Monoid Morphisms; A Reversal-Invariant Decomposition Theory, in preparation. · Zbl 0698.20056
[14] Rhodes, J, Global structure theorems for arbitrary semigroups, (), 197-228
[15] \scJ. Rhodes and B. Tilson, Chapter 8 in [A] entitled, Homomorphism and semilocal theory.
[16] \scG.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrsch.\bf2, 340-368.
[17] Straubing, H, On finite J-trivial monoids, Semigroup forum, 19, 2, 107, (1980) · Zbl 0435.20036
[18] Thue, A, Skr. vid. kristiania, I mat. naturv. klasse., 8, 67, (1912)
[19] \scB. Tilson, Chapters XI and XII in [Ei].
[20] Tilson, B; Tilson, B, The inversion principle with application to decomposition and complexity, J. pure appl. algebra, 4, (1973), Columbia Math Dept, unpublished · Zbl 0293.20049
[21] \scB. Tilson, Categories as Algebra: An Essential Ingredient in the Theory of Monoids, in preparation. · Zbl 0627.20031
[22] \scR. J. Warne, Embedding of regular semigroups in wreath products, J. Pure Appl. Algebra, in press. · Zbl 0572.20045
[23] Murskii, V.L, Example of varieties of semigroups, Mathematical notes of the Academy of sciences of the USSR, Vol. 3, Nos. 5, 6, 423-427, (May-June 1968)
[24] Birkhoff, G, On the structure of abstract algebras, Proc. Cambridge philos. soc., 31, 4, 433-454, (1935) · JFM 61.1026.07
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