Varieties of Regular Pseudocomplemented de Morgan Algebras

In this paper, we investigate the varieties M n and K n of regular pseudocomplemented de Morgan and Kleene algebras of range n , respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in M n and explicitly describe the dual spaces of the simple algebras in M 1 and K 1 . We show that the variety M 1 is locally finite, but this property does not extend to M n or even K n for n =?2. We also show that the lattice of subvarieties of K 1 is an ? +?1 chain and the cardinality of the lattice of subvarieties of either K 2 or M 1 is 2 ? . A description of the lattice of subvarieties of M 1 is given.

[1]  J. A. Kalman,et al.  Lattices with involution , 1958 .

[2]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[3]  Fernando Guzmán,et al.  Subdirectly irreducible and free Kleene-Stone algebras , 1994 .

[4]  H. Priestley,et al.  Distributive Lattices , 2004 .

[5]  Hanamantagouda P. Sankappanavar.  Pseudocomplemented Okham and Demorgan Algebras , 1986, Math. Log. Q..

[6]  H. Priestley.  THE CONSTRUCTION OF SPACES DUAL TO PSEUDOCOMPLEMENTED DISTRIBUTIVE LATTICES , 1975 .

[7]  Anna B. Romanowska.  Subdirectly irreducible pseudocomplemented de Morgan algebras , 1981 .

[8]  J. C. Varlet,et al.  A regular variety of type <2, 2, 1, 1, 0, 0> , 1972 .

[9]  Hanamantagouda P. Sankappanavar,et al.  Congruence properties of pseudocomplemented De Morgan algebras , 2014, Math. Log. Q..

[10]  Hernando Gaitán.  Free Algebras in Certain Varieties of Distributive Pseudocomplemented De Morgan Algebras , 1998, Math. Log. Q..

[11]  William H. Cornish,et al.  Coproducts of Kleene algebras , 1979 .

[12]  B. Jonnson.  Algebras Whose Congruence Lattices are Distributive. , 1967 .

[13]  Algebras of incidence structures: representations of regular double p-algebras , 2016 .

[14]  Hilary A. Priestley,et al.  Ordered Sets and Duality for Distributive Lattices , 1984 .

[15]  Hanamantagouda P. Sankappanavar.  Principal Congruences of Pseudocomplemented Demorgan Algebras , 1987, Math. Log. Q..

[16]  H. P. Sankappanavar,et al.  Regular double p-algebras , 2019, Mathematica Slovaca.

[17]  William H. Cornish,et al.  Coproducts of De Morgan algebras , 1977, Bulletin of the Australian Mathematical Society.

[18]  R. McKenzie,et al.  Algebras, Lattices, Varieties , 1988 .

[19]  Hanamantagouda P. Sankappanavar.  Heyting Algebras with a Dual Lattice Endomorphism , 1987, Math. Log. Q..

[20]  T. Katrinák.  The structure of distributive doublep-algebras. Regularity and congruences , 1973 .

[21]  Hilary A. Priestley,et al.  Representation of Distributive Lattices by means of ordered Stone Spaces , 1970 .

[22]  J. Sichler,et al.  Categorical universality of regular double p-algebras , 1990, Glasgow Mathematical Journal.

[23]  The determination congruence on doublep-algebras , 1976 .

[24]  Xue-Ping Wang,et al.  The Lattices of Kernel Ideals in Pseudocomplemented De Morgan Algebras , 2017, Order.