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We give an example of a finite algebra which is dualizable but not fully dualizable in the sense of natural duality theory.

Every lattice is the complete join of all its one-element sublattices. In this paper we address the question: Which lattices L have the property that L is finitely join reducible in Sub L? That is, when do there exist proper sublattices A, B such that L = A ∨ B? In particular, could it be that every nontrivial lattice has this property, in which case every… (More)

We show that, within the class of three-element unary algebras, there is a tight connection between a finitely based quasi-equational theory, finite rank, enough algebraic operations (from natural duality theory) and a special injectivity condition. A full duality gives us a natural dual equivalence between a quasi-variety generated by a finite algebra and… (More)

Whaley's Theorem on the existence of large proper sublattices of infinite lattices is extended to ordered sets and finite lattices. As a corollary it is shown that every finite lattice L with |L| ≥ 3 contains a proper sublattice S with |S| ≥ |L| 1 3. It is also shown that that every finite modular lattice L with |L| ≥ 3 contains a proper sublattice S with… (More)

Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubsti-tutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvari-eties of a given variety of that type. We then apply the results obtained to the lattice… (More)

We want to extend the duality to semilattices with operators (that preserve + and 0, i.e., endomor-phisms added as operations). For simplicity, let us consider semilattices with one operator: S = S, +, 0, g, as the extension to a monoid of operators is straightforward. 1. Adjoints: the finite case We begin by recalling the general theory of adjoints on… (More)

A finite unary algebra of finite type with a constant function 0 that is a one element subalgebra, and whose operations have range {0, 1}, is called a {0, 1}-valued unary algebra with 0. Such an algebra has a finite basis for its quasi-equations if and only if the relation defined by the rows of the non-trivial functions in the clone form an order ideal.

- D Casperson, J Hyndman, J Mason, J B Nation, B Schaan
- 2015

A finite unary algebra of finite type with a constant function 0 that is a one-element subalgebra, and whose operations have range {0, 1}, is called a {0, 1}-valued unary algebra with 0. Such an algebra has a finite basis for its quasi-equations if and only if the relation defined by the rows of the non-trivial functions in the clone form an order ideal.