Dually quasi-De Morgan Stone semi-Heyting algebras I. Regularity

Document Type: Research Paper

Author

Department of Mathematics, State University of New York, New Paltz, NY 12561

Abstract

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended $lor$-De Morgan law introduced in cite{Sa12}. Then, using this result and the results of cite{Sa12}, we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) in the variety of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. It is shown that there are 25 nontrivial simple algebras in this variety. In Part II, we prove, using the description of simples obtained in this Part, that the variety $mathbf{RDQDStSH_1}$ of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element $mathbf{RDQDStSH_1}$-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras--the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of this theorem, we present (equational) axiomatizations for several subvarieties of $mathbf{RDQDStSH_1}$. The Part II concludes with some open problems for further investigation.

Keywords


[1] M. Abad, J.M. Cornejo and J.P. Diaz Varela, The variety of semi-Heyting algebras
satisfying the equation , Reports on Mathematical Logic
46 (2011), 75-90.
[2] M. Abad, J.M. Cornejo and J.P. Daz Varela, The variety generated by semi-Heyting
chains, Soft Computing 15 (2011), 721-728.
[3] M. Abad and L. Monteiro, Free symmetric Boolean algebras, Revista de la U.M.A.
27 (1976), 207-215.
[4] R. Balbes and PH. Dwinger, Distributive Lattices", Univ. of Missouri Press,
Columbia, 1974.
[5] S. Burris and H.P. Sankappanavar, A Course in Universal Algebra", Springer-
Verlag, New York, 1981. The free, corrected version (2012) is available online as a
PDF file at math.uwaterloo.ca/fisnburris.
[6] G. Gratzer, Lattice Theory", W.H.Freeman and Co., San Francisco, 1971.
[7] A. Horn, Logic with truth values in a linearly ordered Heyting algebras, J. Symbolic.
Logic 34 (1969), 395-408.
[8] B. Jhonsson, Algebras whose congruence lattices are distributive, Math. Scand. 21
(1967), 110-121.
[9] V.Yu. Meskhi, A discriminator variety of Heyting algebras with involution, Algebra
i Logika 21 (1982), 537-552.
[10] A. Monteiro, Sur les algebres de Heyting symetriques, Portugaliae Mathemaica 39
(1980), 1-237.
[11] W. McCune, Prover9 and Mace 4, http://www.cs.unm.edu/mccune/prover9/.
[12] H. Rasiowa, An Algebraic Approach to Non-Classical Logics", North{Holland
Publ.Comp., Amsterdam, 1974.
[13] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics", Warsazawa,
1970.
[14] H.P. Sankappanavar, Heyting algebras with dual pseudocomplementation, Pacific J.
Math. 117 (1985), 405-415.
[15] H.P. Sankappanavar, Pseudocomplemented Okham and De Morgan algebras,
Zeitschr. f. math. Logik und Grundlagen d. Math. 32 (1986), 385-394.
[16] H.P. Sankappanavar, Heyting algebras with a dual lattice endomorphism, Zeitschr.
f. math. Logik und Grundlagen d. Math. 33 (1987), 565{573.
[17] H.P. Sankappanavar, Semi-De Morgan algebras, J. Symbolic. Logic 52 (1987), 712-
724.
[18] H.P. Sankappanavar, Semi-Heyting algebras: An abstraction from Heyting algebras,
Actas del IX Congreso Dr. A. Monteiro (2007), 33-66.
[19] H.P. Sankappanavar, Semi-Heyting algebras II. In Preparation.
[20] H.P. Sankappanavar, Expansions of semi-Heyting algebras. I: Discriminator varieties,
Studia Logica 98 (1-2) (2011), 27-81.
[21] H.P. Sankappanavar, Expansions of semi-Heyting algebras. II. In Preparation.
[22] J, Varlet, A regular variety of type h2; 2; 1; 1; 0; 0i, Algebra Universalis 2 (1972),
218-223.
[23] H. Werner, Discriminator Algebras", Studien zur Algebra und ihre Anwendungen,
Band 6, Academie{Verlag, Berlin, 1978.