On Ockham algebras: Congruence lattices and subdirectly irreducible algebras

Distributive bounded lattices with a dual homomorphism as unary operation, called Ockham algebras, were firstly studied by Berman (1977). The varieties of Boolean algebras, De Morgan algebras, Kleene algebras and Stone algebras are some of the well known subvarieties of Ockham algebra. In this paper, new results about the congruence lattice of Ockham algebras are given. From these results and Urquhart's representation theorem for Ockham algebras a complete characterization of the subdirectly irreducible Ockham algebras is obtained. These results are particularized for a large number of subvarieties of Ockham algebras. For these subvarieties a full description of their subdirectly irreducible algebras is given as well.

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

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

[3]  Representation theorem of Ockham algebras , 1989, Proceedings. The Nineteenth International Symposium on Multiple-Valued Logic.

[4]  G. Grätzer.  General Lattice Theory , 1978 .

[5]  W. Blok,et al.  On the structure of varieties with equationally definable principal congruences IV , 1994 .

[6]  Andrzej Białynicki-Birula,et al.  Remarks on quasi-Boolean algebras , 1957 .

[7]  H. Gericke.  Białynicki–Birula A.. Remarks on quasi–Boolean algebras. Bulletin de l′Académie Polonaise des Sciences, Classe III, Bd. 5 (1957), S. 615–619. , 1967 .

[8]  H. Gericke.  Białynicki-Birula A. and Rasiowa H.. On the representation of quasi-Boolean algebras. Bulletin de l'Académie Polonaise des Sciences , Classe III, Bd. 5 (1957), S. 259–261. , 1957 .

[9]  Roberto Cignoli.  Injective De Morgan and Kleene algebras , 1975 .

[10]  P. Köhler,et al.  Varieties with equationally definable principal congruences , 1980 .

[11]  G. Grätzer,et al.  Ideals and congruence relations in lattices , 1958 .

[12]  Joel Berman,et al.  Distributive lattices with an additional unary operation , 1977 .

[13]  Helena Rasiowa,et al.  On the Representation of Quasi-Boolean Algebras , 1957 .

[14]  George Gratzer,et al.  Universal Algebra , 1979 .

[15]  Hanamantagouda P. Sankappanavar.  Distributive Lattices with a Dual Endomorphism , 1985, Math. Log. Q..

[16]  Harry Lakser,et al.  Principal congruences of pseudocomplemented distributive lattices , 1973 .

[17]  Representation of symmetric algebras and its subvarieties , 1988, [1988] Proceedings. The Eighteenth International Symposium on Multiple-Valued Logic.

[18]  H. P. Sankappanavar.  A Characterization of Principal Congruences of De Morgan Algebras and its Applications , 1980 .

[19]  Alasdair Urquhart,et al.  Distributive lattices with a dual homomorphic operation , 1979 .