On some congruences of power algebras

Agata Pilitowska; Anna Zamojska-Dzienio

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 987-1003
  • ISSN: 2391-5455

Abstract

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In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).

How to cite

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Agata Pilitowska, and Anna Zamojska-Dzienio. "On some congruences of power algebras." Open Mathematics 10.3 (2012): 987-1003. <http://eudml.org/doc/269105>.

@article{AgataPilitowska2012,
abstract = {In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).},
author = {Agata Pilitowska, Anna Zamojska-Dzienio},
journal = {Open Mathematics},
keywords = {Idempotent; Entropic; Modes; Power algebras; Congruence relations; Identities; Finitary algebras; idempotent algebra; entropic algebra; mode; modal; power algebra; congruence relation; identity; finitary algebra; idempotent quotient},
language = {eng},
number = {3},
pages = {987-1003},
title = {On some congruences of power algebras},
url = {http://eudml.org/doc/269105},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Agata Pilitowska
AU - Anna Zamojska-Dzienio
TI - On some congruences of power algebras
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 987
EP - 1003
AB - In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).
LA - eng
KW - Idempotent; Entropic; Modes; Power algebras; Congruence relations; Identities; Finitary algebras; idempotent algebra; entropic algebra; mode; modal; power algebra; congruence relation; identity; finitary algebra; idempotent quotient
UR - http://eudml.org/doc/269105
ER -

References

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  1. [1] Adaricheva K., Pilitowska A., Stanovský D., On complex algebras of subalgebras, Algebra Logika, 2008, 47(6), 655–686 (in Russian) http://dx.doi.org/10.1007/s10469-008-9036-7 Zbl1241.08002
  2. [2] Bošnjak I., Madarász R., On power structures, Algebra Discrete Math., 2003, 2, 14–35 Zbl1063.08001
  3. [3] Brink C., Power structures, Algebra Universalis, 1993, 30(2), 177–216 http://dx.doi.org/10.1007/BF01196091 
  4. [4] Ježek J., Markovic P., Maróti M., McKenzie R., The variety generated by tournaments, Acta Univ. Carolin. Math. Phys., 1999, 40(1), 21–41 Zbl0939.08003
  5. [5] Ježek J., McKenzie R., The variety generated by equivalence algebras, Algebra Universalis, 2001, 45(2–3), 211–219 http://dx.doi.org/10.1007/s000120050213 Zbl0980.08004
  6. [6] Jónsson B., Tarski A., Boolean algebras with operators. I, Amer. J. Math., 1951, 73, 891–939 http://dx.doi.org/10.2307/2372123 Zbl0045.31505
  7. [7] Jónsson B., Tarski A., Boolean algebras with operators. II, Amer. J. Math., 1952, 74, 127–162 http://dx.doi.org/10.2307/2372074 Zbl0045.31601
  8. [8] Kearnes K., Semilattice modes I: the associated semiring, Algebra Universalis, 1995, 34(2), 220–272 http://dx.doi.org/10.1007/BF01204784 Zbl0848.08005
  9. [9] McKenzie R.N., McNulty G., Taylor W., Algebras, Lattices, Varieties. I, Wadsworth Brooks/Cole Math. Ser., Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, 1987 Zbl0611.08001
  10. [10] McKenzie R., Romanowska A., Varieties of ·-distributive bisemilattices, Contributions to General Algebra, Klagenfurt, May 25–28, 1978, Heyn, Klagenfurt, 1979, 213–218 
  11. [11] Pilitowska A., Modes of Submodes, PhD thesis, Warsaw University of Technology, 1996 
  12. [12] Pilitowska A., Zamojska-Dzienio A., Representation of modals, Demonstratio Math., 2011, 44(3), 535–556 Zbl1239.08003
  13. [13] Pilitowska A., Zamojska-Dzienio A., Varieties generated by modes of submodes, preprint available at http://www.mini.pw.edu.pl/~apili/publikacje.html Zbl1270.08004
  14. [14] Romanowska A., Semi-affine modes and modals, Sci. Math. Jpn., 2005, 61(1), 159–194 Zbl1067.08001
  15. [15] Romanowska A.B., Roszkowska B., On some groupoid modes, Demonstratio Math., 1987, 20(1–2), 277–290 Zbl0669.08005
  16. [16] Romanowska A.B., Smith J.D.H., Bisemilattices of subsemilattices, J. Algebra, 1981, 70(1), 78–88 http://dx.doi.org/10.1016/0021-8693(81)90244-1 Zbl0457.06002
  17. [17] Romanowska A.B., Smith J.D.H., Modal Theory, Res. Exp. Math., 9, Heldermann, Berlin, 1985 
  18. [18] Romanowska A.B., Smith J.D.H., Subalgebra systems of idempotent entropic algebras, J. Algebra, 1989, 120(2), 247–262 http://dx.doi.org/10.1016/0021-8693(89)90197-X 
  19. [19] Romanowska A.B., Smith J.D.H., On the structure of subalgebra systems of idempotent entropic algebras, J. Algebra, 1989, 120(2), 263–283 http://dx.doi.org/10.1016/0021-8693(89)90198-1 
  20. [20] Romanowska A.B., Smith J.D.H., Modes, World Scientific, River Edge, 2002 

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