Congruence properties of pseudocomplemented De Morgan algebras

In this paper we first describe the Priestley duality for pseudocomplemented De Morgan algebras by combining the known dualities of distributive p‐algebras due to Priestley and for De Morgan algebras due to Cornish and Fowler. We then use it to characterize congruence‐permutability, principal join property, and the property of having only principal congruences for pseudocomplemented De Morgan algebras. The congruence‐uniform pseudocomplemented De Morgan algebras are also described.     

A note on chain-based semi-Heyting algebras

We determine the number of non-isomorphic semi-Heyting algebras on an n-element chain, where n is a positive integer, using a recursive method. We then prove that the numbers obtained agree with those determined in [1]. We apply the formula to calculate the number of non-isomorphic semi-Heyting chains of a given size in some important subvarieties of the variety of semi-Heyting algebras that were introduced in [5]. We further exploit this recursive method to calculate the numbers $A(n,m)$ of non-isomorphic semi-Heyting chains with n elements such that removing the mth element ($1<m<n$) we are left with a subalgebra. We also solve a related problem posed in [1] of determining the number of ways a semi-Heyting chain with $n-1$ elements can be extended to a n element semi-Heyting chain by adding a new element in the mth place. Finally we combine these results by finding a second way to calculate the numbers $A(n,m)$ that provides some extra information.     

Principal congruences of double demi-p-lattices

A formula is given to express a principal congruence on a double demi-p-lattice as a join of countably many principal lattice congruences. It is then applied to show that the variety of double demi-p-lattices has the congruence extension property. As special cases one obtains some known results for distributive doublep-lattices due to T. Hecht and T. Katriák.Presented by Bjarni Jónsson.     

Varieties of Regular Pseudocomplemented de Morgan Algebras

Order - In this paper, we investigate the varieties Mn and Kn of regular pseudocomplemented de Morgan and Kleene algebras of range n, respectively. Priestley duality as it applies to...     

Quasi-Stone algebras

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras (QSA's). Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is locally finite. Furthermore we prove that the lattice of subvarieties of QSA's is an (ω + 1)-chain. MSC: 03G25, 06D16, 06E15.     

Universal varieties of quasi-Stone algebras

In this paper, we are mainly concerned with semicontinuity of complete lattices and their distributive reflections, introduced by Rav in 1989. We prove that for a complete lattice L, the distributive reflection L_d is isomorphic to the lattice of all radicals determined by principal ideals of L in the set-inclusion order, obtaining a method to depict the distributive reflection of a given lattice. It is also proved that if a complete lattice L is semicontinuous and every semiprime element x ∈ L is the largest in d(x), then L_d is continuous whenever the distributive reflector d is Scott continuous. We construct counterexamples to confirm a conjecture and solve two open problems posed by Zhao in 1997.