On some congruences of power algebras

• Autorzy: Agata Pilitowska , Anna Zamojska-Dzienio
• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

Idempotent; Entropic; Modes; Power algebras; Congruence relations; Identities; Finitary algebras

Kategorie tematyczne

• 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.)
• 08A62: Finitary algebras
• 08B99: None of the above, but in this section
• 06A12: Semilattices
• 08A30: Subalgebras, congruence relations
• 08A99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 3
• Strony: 987-1003

Daty

wydano

2012-06-01

online

2012-03-24

Twórcy

autor

Agata Pilitowska

• Warsaw University of Technology

autor

Anna Zamojska-Dzienio

• Warsaw University of Technology

Bibliografia

• [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
• [2] Bošnjak I., Madarász R., On power structures, Algebra Discrete Math., 2003, 2, 14–35
• [3] Brink C., Power structures, Algebra Universalis, 1993, 30(2), 177–216 http://dx.doi.org/10.1007/BF01196091
• [4] Ježek J., Markovic P., Maróti M., McKenzie R., The variety generated by tournaments, Acta Univ. Carolin. Math. Phys., 1999, 40(1), 21–41
• [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
• [6] Jónsson B., Tarski A., Boolean algebras with operators. I, Amer. J. Math., 1951, 73, 891–939 http://dx.doi.org/10.2307/2372123
• [7] Jónsson B., Tarski A., Boolean algebras with operators. II, Amer. J. Math., 1952, 74, 127–162 http://dx.doi.org/10.2307/2372074
• [8] Kearnes K., Semilattice modes I: the associated semiring, Algebra Universalis, 1995, 34(2), 220–272 http://dx.doi.org/10.1007/BF01204784
• [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
• [10] McKenzie R., Romanowska A., Varieties of ·-distributive bisemilattices, Contributions to General Algebra, Klagenfurt, May 25–28, 1978, Heyn, Klagenfurt, 1979, 213–218
• [11] Pilitowska A., Modes of Submodes, PhD thesis, Warsaw University of Technology, 1996
• [12] Pilitowska A., Zamojska-Dzienio A., Representation of modals, Demonstratio Math., 2011, 44(3), 535–556
• [13] Pilitowska A., Zamojska-Dzienio A., Varieties generated by modes of submodes, preprint available at http://www.mini.pw.edu.pl/~apili/publikacje.html
• [14] Romanowska A., Semi-affine modes and modals, Sci. Math. Jpn., 2005, 61(1), 159–194
• [15] Romanowska A.B., Roszkowska B., On some groupoid modes, Demonstratio Math., 1987, 20(1–2), 277–290
• [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
• [17] Romanowska A.B., Smith J.D.H., Modal Theory, Res. Exp. Math., 9, Heldermann, Berlin, 1985
• [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] 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] Romanowska A.B., Smith J.D.H., Modes, World Scientific, River Edge, 2002

Identyfikatory

DOI

10.2478/s11533-012-0018-6