zbMATH — the first resource for mathematics

Prime decomposition theorem for arbitrary semigroups: General holonomy decomposition and synthesis theorem. (English) Zbl 0679.20056
The authors generalize the holonomy form of the Prime Decomposition Theorem of Krohn and Rhodes for finite semigroups to arbitrary infinite semigroups. This is accomplished by embedding \(\hat S\) into an infinite Zeiger wreath product after applying the triple Schützenberger product which makes S finite-J-above (Rhodes’ theorem). Here finite-J-above means every element has only a finite number of divisors.
Reviewer: L.Potemkin

20M10 General structure theory for semigroups
20M50 Connections of semigroups with homological algebra and category theory
20M05 Free semigroups, generators and relations, word problems
Full Text: DOI
[1] Algebraic theory of machines, () · Zbl 0138.00808
[2] Birget, J.C., The synthesis theorem for finite regular semigroups and its generalization, J. pure appl. algebra, 55, 1-79, (1988), this issue. · Zbl 0657.20050
[3] Birget, J.C., Structure of finite semigroups and generalizations, () · Zbl 1328.68072
[4] Birget, J.C., Iteration of expansions - unambiguous semigroups, J. pure appl. algebra, 34, 1-55, (1984) · Zbl 0547.20054
[5] Birget, J.C.; Rhodes, J.L., Almost finite expansions of arbitrary semigroups, J. pure appl. algebra, 32, 239-287, (1984) · Zbl 0546.20055
[6] Clifford, A.H.; Preston, G.B., The algebraic theory of semigroups, vol. I and II, (1967), American Mathematical Society Providence, RI · Zbl 0178.01203
[7] Eilenberg, S., Automata, languages and machines, vol. B, (1976), Academic Press New York
[8] Henckell, K., Pointlike sets: the finest aperiodic cover of a finite semigroup, J. pure appl. algebra, 55, 85-126, (1988), this issue. · Zbl 0682.20044
[9] Lallement, G., Semigroups and combinatorial applications, (1979), Wiley New York · Zbl 0421.20025
[10] Pin, J.E., Varieties of formal languages, (1984), Plenum New York · Zbl 0655.68095
[11] Rhodes, J., Infinite iteration of matrix semigroups, part I: structure theorem for torsion semig-groups, J. algebra, 98, 422-451, (1986) · Zbl 0584.20053
[12] Rhodes, J., Infinite iteration of matrix semigroups, part II: structure theorem for arbitrary semi-groups up to aperiodic morphism, J. algebra, 100, 25-137, (1986) · Zbl 0626.20050
[13] Rhodes, J.; Allen, D., Synthesis of the classical and modern theory of finite semigroups, Adv. math., 11, 238-266, (1973)
[14] J. Rhodes and B. Tilson, The kernel of monoid morphisms, to be published. · Zbl 0698.20056
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.