# Dually quasi-De Morgan Stone semi-Heyting algebras II. Regularity

Document Type: Research Paper

Author

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

Abstract

This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, 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 our main theorem, we present (equational) axiomatizations for several subvarieties of $mathbf{RDQDStSH_1}$. The paper concludes with some open problems for further investigation.

Keywords

### References

[1] R. Balbes and PH. Dwinger, Distributive Lattices", Univ. of Missouri Press,
Columbia, 1974.

[2] 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/snburris.
[3] B. Jhonsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121.
[4] W. McCune, Prover9 and Mace 4, http://www.cs.unm.edu/mccune/prover9/.
[5] H. Rasiowa, An Algebraic Approach to Non-Classical Logics", North{Holland Publ.Comp., Amsterdam, 1974.
[6] H.P. Sankappanavar, Heyting algebras with dual pseudocomplementation, Pacific J. Math. 117 (1985), 405-415.
[7] H.P. Sankappanavar, Heyting algebras with a dual lattice endomorphism, Zeitschr. f. math. Logik und Grundlagen d. Math. 33 (1987), 565{573.
[8] H.P. Sankappanavar, Semi-De Morgan algebras, J. Symbolic. Logic 52 (1987), 712- 724.
[9] H.P. Sankappanavar, Semi-Heyting algebras: An abstraction from Heyting algebras, Actas del IX Congreso Dr. A. Monteiro (2007), 33-66.
[10] H.P. Sankappanavar, Expansions of semi-Heyting algebras. I: Discriminator varieties, Studia Logica 98 (1-2) (2011), 27-81.
[11] H.P. Sankappanavar, Dually quasi-De Morgan Stone semi-Heyting Regularity, Categ. General Alg. Struct. Appl. 2(1) (2014), 47-64.