Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines
01 Apr 1965-Transactions of the American Mathematical Society (American Mathematical Society (AMS))-Vol. 116, pp 450-464
TL;DR: In this paper, it was shown that a semigroup S is irreducible if and only if either: either: (1) S I S2 or (2)S I S1.
Abstract: Introduction. In the following all semigroups are of finite order. One semigroup S, is said to divide another semigroup S2, written SlIS2, if S, is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Sl, with connecting homomorphism Y, is written S2 X y Sl. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S I (S2 X Y SJ) implies S I S2 or S I S1. It is shown that S is irreducible if and only if either:
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TL;DR: In this paper, a new ordering for categories, called division, is introduced which allows useful comparison of monoids and categories, and a strong connection is established between category varieties and certain semigroup varieties associated with recognizable languages.
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TL;DR: The use of two-sided quotients adds insight to the theory of star-free events and permits the derivation of some new properties of these events; in particular, every star- free event has at least one quotient which is either empty or full.
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TL;DR: In this paper, a review of multivariable feedback system design techniques from the frequency-response viewpoint is given, including a comparison of the advantages and disadvantages of each type of design method, including Non-Interacting Control, Modal Control, Optimal Control, Commutative Control, The Inverse Nyquist Array and The Characteristic Locus.
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TL;DR: These automata groups are similar in nature to lamplighters: they are amenable locally-finite-by-cyclic groups; under mild decidable conditions, the semigroups generated by such automata are free.
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100 citations
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Book•
01 Jan 1964
TL;DR: A survey of the structure and representation theory of semi groups is given in this article, along with an extended treatment of the more important recent developments of Semi Group Structure and Representation.
Abstract: This book, along with volume I, which appeared previously, presents a survey of the structure and representation theory of semi groups. Volume II goes more deeply than was possible in volume I into the theories of minimal ideals in a semi group, inverse semi groups, simple semi groups, congruences on a semi group, and the embedding of a semi group in a group. Among the more important recent developments of which an extended treatment is presented are B. M. Sain's theory of the representations of an arbitrary semi group by partial one-to-one transformations of a set, L. Redei's theory of finitely generated commutative semi groups, J. M. Howie's theory of amalgamated free products of semi groups, and E. J. Tully's theory of representations of a semi group by transformations of a set. Also a full account is given of Malcev's theory of the congruences on a full transformation semi group.
3,533 citations
Book•
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01 Jan 1959
TL;DR: The theory of normal subgroups and homomorphisms was introduced in this article, along with the theory of $p$-groups regular $p-groups and their relation to abelian groups.
Abstract: Introduction Normal subgroups and homomorphisms Elementary theory of abelian groups Sylow theorems Permutation groups Automorphisms Free groups Lattices and composition series A theorem of Frobenius solvable groups Supersolvable and nilpotent groups Basic commutators The theory of $p$-groups regular $p$-groups Further theory of abelian groups Monomial representations and the transfer Group extensions and cohomology of groups Group representation Free and amalgamated products The Burnside problem Lattices of subgroups Group theory and projective planes Bibliography Index Index of special symbols.
2,960 citations
TL;DR: Finite automata are considered as instruments for classifying finite tapes as well as generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined.
Abstract: Finite automata are considered in this paper as instruments for classifying finite tapes. Each one-tape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms.
1,930 citations
"Algebraic theory of machines. I. Pr..." refers background in this paper
...(2) See references [3], [4], [6], and [7], and §4 of this paper for a discussion of machines and automata....
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TL;DR: The homomorphism and equivalence of automata and semi-groups are studied as well as operations on events in finite automata, and experiments with automata are conducted.
Abstract: CONTENTSIntroduction § 1. Homomorphism and equivalence of automata § 2. Introduction of mappings in automata § 3. Introduction of events in finite automata, operations on events § 4. Automata and semi-groups § 5. The composition of automata § 6. Experiments with automataConclusionReferences
475 citations