Ockham algebras with De Morgan skeletons. (English) Zbl 0671.06007
An Ockham algebra (L,f) consists of a bounded distributive lattice L and of a dual endomorphism f of L. The authors deal with the class \(K_{1,1}\) of Ockham algebras defined by \(f=f^ 3\). There are 19 non- trivial subdirectly irreducible algebras in \(K_{1,1}\) (H. Sankanappanavar and R. Beazer). The main result of the paper is the description of the lattice of all subvarieties of \(K_{1,1}\); see Section 2: this lattice has 403 elements and a rather complicated structure; an impressive result.
In section 4, finite equational bases are provided for many of these varieties.
In section 4, finite equational bases are provided for many of these varieties.
Reviewer: G.Grätzer
MSC:
| 06D99 | Distributive lattices |
Keywords:
lattice of subvarieties; Ockham algebra; bounded distributive lattice; dual endomorphism; finite equational basesReferences:
| [1] | Beazer, R., On some small varieties of distributive Ockham algebras, Glasgow Math. J., 25, 175-181 (1984) · Zbl 0539.06012 |
| [2] | Berman, J., Distributive lattices with an additional unary operation, Aequationes Math., 16, 165-171 (1977) · Zbl 0395.06007 |
| [3] | Berman, J.; Köhler, P., Finite distributive lattices and finite partially ordered sets, (Mitt. Math. Sem. Giessen, 121 (1976)), 103-124 · Zbl 0355.06015 |
| [4] | Blyth, T. S.; Varlet, J. C., On a common abstraction of de Morgan algebras and Stone algebras, (Proc. Roy. Soc. Edinburgh Sect. A, 94 (1983)), 301-308 · Zbl 0536.06013 |
| [5] | Blyth, T. S.; Varlet, J. C., Subvarieties of the class of MS-algebras, (Proc. Roy. Soc. Edinburgh Sect. A, 95 (1983)), 157-169 · Zbl 0544.06011 |
| [6] | Davey, B. A., On the lattice of subvarieties, Houston J. Math., 5, 183-192 (1979) · Zbl 0396.08008 |
| [7] | Goldberg, M. S., Distributive \(p\)-Algebras and Ockham Algebras, a Topological Approach, (Ph. D. thesis (1979), La Trobe University: La Trobe University Australia) · Zbl 0427.06004 |
| [8] | Sankappanavar, H., Distributive lattices with a dual endomorphism, Z. Math. Logik Grundlag. Math., 31, 385-392 (1985) · Zbl 0578.06010 |
| [9] | Urquhart, A., Distributive lattices with a dual homomorphic operation, Studia Logica, 40, 391-404 (1981) · Zbl 0511.06007 |
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