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Random semigroup acts on a finite set

Published online by Cambridge University Press:  09 April 2009

Göran Högnäs
Göran Högnäs
Affiliation:
Department of Mathematics, Åbo Akademi, Henriksgatan 9, SF-20500 Åbo 50, Finland.
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Abstract

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Let X be a finite set and S a semigroup of transformations of X. We investigate the trace on X of a random walk on S. We relate the structure of the trace process, which turns out to be a Markov chain, to that of the random walk. We show, for example, that all periods of the trace process divide the period of the random walk.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 4 , June 1977 , pp. 481 - 498
Copyright
Copyright © Australian Mathematical Society 1977

References

Arbib, M. (ed.) (1968), Algebraic Theory of Machines, Languages, and Semigroups (Academic Press, 1968).Google Scholar
Clifford, A. H. and Preston, G. B. (1961), The Algebraic Theory of Semigroups, Volume I (Amer. Math. Soc., 1961).Google Scholar
Day, J. M. and Wallace, A. D. (1967), ‘Multiplication induced in the state space of an act’, Math. Sys. Theory 1, 305–314.CrossRefGoogle Scholar
Deussen, P. (1971), Halbgruppen und Automaten (Springer, 1971).CrossRefGoogle Scholar
Grenander, U. (1963), Probabilities on Algebraic Structures (Wiley, 1963).Google Scholar
Hall, M. (1959), The Theory of Groups (MacMillan, 1959).Google Scholar
Högnäs, G. (1974a), ‘Marches aléatoires sur un demi-groupe compact’, Ann. Inst. Henri Poincaré, Sect. B 10, 115–154.Google Scholar
Högnäs, G. (1974b), ‘Remarques sur les marches aléatoires dans un demi-groupe avec un idéal compact ayant une probabilité positive’, Ann. Inst. Henri Poincaré, Sect. B 10, 345–354.Google Scholar
Kemeny, J. K., Snell, J. L., and Knapp, A. W. (1966), Denumerable Markov Chains (Van Nostrand, 1966).Google Scholar
Larisse, J. (1972), ‘Marches au hasard sur les demi-groupes discrets’, Ann. Inst. Henri Poincaré, Sect. B 8, 107–125, 127–173, 229–240.Google Scholar
Martin-Löf, P. (1965), ‘Probability theory on discrete semigroups’, Z. Wahrscheinlichkeitstheorie 4, 78–102.CrossRefGoogle Scholar
Mukherjea, A., Sun, T. C. and Tserpes, N. A. (1973), ‘Random walks on compact semigroups’, Proc. Amer. Math. Soc. 39, 599–609.CrossRefGoogle Scholar
Rosenblatt, M. (1971), Markov Processes. Structure and Asymptotic Behavior (Springer, 1971).Google Scholar
Schwarz, S. (1964), ‘Convolution semigroup of measures on compact non-commutative semi-groups’, Czechosl. Math. J. 14 (89), 95–115.CrossRefGoogle Scholar
Stadtlander, D. (1968), ‘Thread actions’, Duke Math. J. 35, 483–490.CrossRefGoogle Scholar
Sun, T. C., Mukherjea, A. and Tserpes, N. A. (1973), ‘On recurrent random walks on semigroups’, Trans. Amer. Math. Soc. 185, 213–227.CrossRefGoogle Scholar
Wallace, A. D. (1957), ‘Retractions in semigroups’, Pac. J. Math. 7, 1513–1517.CrossRefGoogle Scholar
Wallace, A. D. (1963), ‘Relative ideals in semigroups. II’, Acta Math. Acad. Sci. Hung. 14, 137–148.CrossRefGoogle Scholar