Congruence Pairs of Decomposable MS-Algebras

Abstract

In this paper, the authors first introduce the concept of congruence pairs on the class of decomposable MS-algebras generalizing that for principal MS-algebras (see [13]). They show that every congruence relation θ on a decomposable MS-algebra L can be uniquely determined by a congruence pair (θ ₁ ,θ ₂), where θ ₁ is a congruence on the de Morgan subalgebra L°° of L and θ ₂ is a lattice congruence on the sublattice D(L) of L. They obtain certain congruence pairs of a decomposable MS-algebra L via central elements of L. Moreover, they characterize the permutability of congruences and the strong extensions of decomposable MS-algebras in terms of congruence pairs.

Keywords

MS-Algebras / Decomposable MS-algebras / Congruence pairs / Strong extension / Permutability / Congruence lattices

References

[1] Badawy A. d _L-Filters of principal MS-algebras. Journal of the Egyptian Mathematical Society, 2015, 23: 463-469

[2] Badawy A. Regular double MS-algebras. Appl. Math. Inf. Sci., 2017, 11(2): 115-122

[3] Badawy A. Characterization of the congruence lattices of principal p-algebras. Mathematica Slovaca, 2017, 67: 803-810

[4] Badawy A. Constructions of core regular double MS-algebras. Filomat, 2018, 34(1): 35-50

[5] Badawy A. Extensions of the Gilvenko-type congruences on a Stone lattice. Math. Meth. Appl. Sci., 2018, 41: 5719-5732

[6] Badawy A. Balanced factor congruences of double MS-algebras. Journal of the Egyptian Mathematical Society, 2019, 27(6): 1-15

[7] Badawy A. Congruences and de Morgan filters of decomposable MS-algebras. Southeast Asian Bulletin of Mathematics, 2019, 43: 13-25

[8] Badawy A, Atallah M. MS-intrvals of an MS-algebra. Hacet. J. Math. Stat., 2019, 48(4): 1479-1487

[9] Badawy A, El-Fawal R. Homomorphism and subalgebras of decomposable MS-algebras. Journal of Egyptian Mathematical Society, 2017, 25: 119-124

[10] Badawy A, El-Fawal R. Closure filters of decomposable MS-algebras. Southeast Asian Bulletin of Mathematics, 2020, 44: 177-194

[11] Badawy A, Gaber A. Complete decomposable MS-algebras. Journal of the Egyptian Mathematical Society, 2019, 27(23): 1-11

[12] Badawy A, Guffova D, Haviar M. Triple construction of decomposable MS-algebras. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica, 2012, 51(2): 53-65

[13] Badawy A, Haviar M, Ploščica M. Congruence pairs of principal MS-algebras and perfect extensions. Mathematica Slovaca, 2020, 70(6): 1275-1288

[14] Badawy A, Hussen S, Gaber A. Quadruple construction of decomposable double MS-algebras. Mathematica Slovaca, 2020, 70(5): 1041-1056

[15] Badawy A, Sambasiva Rao M. Closure ideas of MS-algebras. Chamchuri J. Math., 2014, 6: 31-46

[16] Badawy A, Shum K P. Congruence pairs of principal p-algebras. Mathematica Slovaca, 2017, 67: 263-270

[17] Beazer R. Congruence pairs for algebras abstracting Kleene algebras and Stone algebras. Czechoslovak Math. J., 1985, 35: 260-268

[18] Blyth T S, Varlet J C. On a common abstraction of de Morgan algebras and Stone algebras. Proc. Roy. Soc. Edinburgh, 1983, 94A: 301-308

[19] Blyth T S, Varlet J C. Subvarieties of the class of MS-algebras. Proc. Roy. Soc. Edinburgh, 1983, 95A: 157-169

[20] Blyth T S, Varlet J C. Ockham algebras, 1994, New York: Oxford Science Publications, The Clarendon Press, Oxford University Press

[21] El-Assar S. Two notes of the congruence lattices of p-algebras. Acta Math. Univ. Comenian., 1985, XLVI-XLVII: 13-19

[22] El-Assar S. Strong extensions of quasi-modular p-algebras. Delta J. Sci., 1988, 12: 843-851

[23] El-Assar S, Badawy A. Homomorphisms and Subalgebras of MS-algebras. Quter Univ. Sci. J., 1995, 15(2): 279-289

[24] El-Assar S, Abdel-Hakim M. Two notes on modular p-algebras. Qutar Univ. Sci. Bull., 1988, 8: 23-28

[25] Gaber A, Badawy A, Hussen S. On decomposable MS-algebras. Italian J. of Pure and Appl. Math., 2020, 43: 617-626

[26] Grätzer G. Lattice theory, first concepts and distributive lattices, 1971, San Francisco, California: Freeman

[27] Katriñák T. On a problem of G. Grätzer. Proc. Amer. Math. Soc., 1976, 57(1): 19-24

[28] Katriñák T. Essential and strong extensions of p-algebras. Bull. de la Soc. Roy. des Sci. de Liege, 1980, 3-4: 119-124

[29] Katriñák T. p-Algebras, Contributions to Lattice Theory, Szeged 1980 Colloq. Math. Soc Janos Bolyai, 1983, Amsterden: North-Holland 549-573 33

[30] Luo C W. The congruence extension property of MS-algebras. J. Wuhan Univ. Natur. Sci. Ed., 1998, 44(3): 282-284

[31] Rao M S, Badawy A. Filters of lattices withrespect to a congruence, Discussions Mathematics. General Algebra and Applications, 2014, 34: 211-217

[32] Varlet J C. A strengthening of the notion of essential extension. Bull. de la Soc. Roy. des Sci. de Liege, 1979, 11-12: 432-437