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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)

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.

Technology is at a stage where it has infiltrated the education system with the potential to enhance teaching and learning. In Northern Ireland a Virtual Learning Environment (VLE) infrastructure is in place. However, statistics and government reports suggest that VLE use amongst the primary school sector is quite limited. In an attempt to redress the… (More)

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