A class of quasigroups solving a problem of ergodic theory

Jonathan D. H. Smith

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 2, page 409-414
  • ISSN: 0010-2628

Abstract

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A pointed quasigroup is said to be semicentral if it is principally isotopic to a group via a permutation on one side and a group automorphism on the other. Convex combinations of permutation matrices given by the one-sided multiplications in a semicentral quasigroup then yield doubly stochastic transition matrices of finite Markov chains in which the entropic behaviour at any time is independent of the initial state.

How to cite

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Smith, Jonathan D. H.. "A class of quasigroups solving a problem of ergodic theory." Commentationes Mathematicae Universitatis Carolinae 41.2 (2000): 409-414. <http://eudml.org/doc/248645>.

@article{Smith2000,
abstract = {A pointed quasigroup is said to be semicentral if it is principally isotopic to a group via a permutation on one side and a group automorphism on the other. Convex combinations of permutation matrices given by the one-sided multiplications in a semicentral quasigroup then yield doubly stochastic transition matrices of finite Markov chains in which the entropic behaviour at any time is independent of the initial state.},
author = {Smith, Jonathan D. H.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; Latin square; Markov chain; doubly stochastic matrix; ergodic; superergodic; dripping faucet; group isotope; central quasigroup; semicentral quasigroup; $T$-quasigroup; left linear quasigroup; pointed quasigroups; Latin squares; Markov chains; doubly stochastic matrices; ergodicity; superergodicity; group isotopes; semicentral quasigroups; left linear quasigroups},
language = {eng},
number = {2},
pages = {409-414},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A class of quasigroups solving a problem of ergodic theory},
url = {http://eudml.org/doc/248645},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Smith, Jonathan D. H.
TI - A class of quasigroups solving a problem of ergodic theory
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 2
SP - 409
EP - 414
AB - A pointed quasigroup is said to be semicentral if it is principally isotopic to a group via a permutation on one side and a group automorphism on the other. Convex combinations of permutation matrices given by the one-sided multiplications in a semicentral quasigroup then yield doubly stochastic transition matrices of finite Markov chains in which the entropic behaviour at any time is independent of the initial state.
LA - eng
KW - quasigroup; Latin square; Markov chain; doubly stochastic matrix; ergodic; superergodic; dripping faucet; group isotope; central quasigroup; semicentral quasigroup; $T$-quasigroup; left linear quasigroup; pointed quasigroups; Latin squares; Markov chains; doubly stochastic matrices; ergodicity; superergodicity; group isotopes; semicentral quasigroups; left linear quasigroups
UR - http://eudml.org/doc/248645
ER -

References

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  2. Belyavskaja G.B., Tabarov A.H., The nuclei and center of linear quasigroups (in Russian), Izv. Akad. Nauk Respub. Moldova Mat. 3 (1991), 37-42. (1991) MR1174875
  3. Belyavskaja G.B., Tabarov A.H., One-sided T-quasigroups and irreducible balanced identities, Quasigroups Related Systems 1 (1994), 8-21. (1994) MR1327942
  4. Chein O., Pflugfelder H.O., Smith J.D.H. (eds.), Quasigroups and Loops: Theory and Applications, Heldermann Berlin (1990). (1990) Zbl0719.20036MR1125806
  5. Feller W., An Introduction to Probability Theory and its Applications, Volume I, Wiley New York, NY (1950). (1950) MR0038583
  6. Horibe Y., On the increase of conditional entropy in Markov chains, in ``Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Volume A'', Academia, Prague, 1988, pp.391-396. Zbl0707.60059MR1136296
  7. Ježek J., Kepka T., Quasigroups, isotopic to a group, Comment. Math. Univ. Carolinae 16 (1975), 59-76. (1975) MR0367103
  8. Němec P., Kepka T., T-quasigroups I, II, Acta Univ. Carolinae - Math. et Phys. 12 (1971), 1 39-49 and no. 2, 31-49. (1971) MR0320206
  9. Smith J.D.H., Mal'cev Varieties, Springer Berlin (1976). (1976) Zbl0344.08002MR0432511
  10. Smith J.D.H., Romanowska A.B., Post-Modern Algebra, Wiley New York, NY (1999). (1999) Zbl0946.00001MR1673047

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