Contributions to the theory of Ockham algebras
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In the first part of this thesis we consider particular ordered sets (connected and of small height) and determine the cardinality of the corresponding dual MS - algebra and of its set of fixed points. The remainder of the thesis is devoted to a study of congruences of Ockham algebras and a generalised variety K𝜔 of Ockham algebras that contains all of the Berman varieties K[sub]p,[sub]q. In particular we consider the congruences [sub]i(i = 1, 2,...) defined on an Ockham algebra (L; f) by (x, y) ∊ [sub]i ⇔ fⁱ(x)= fⁱ(y) and show that (L; f) ∊ K𝜔 is subdirectly irreducible if and only if the lattice of congruences of L reduces to the chain 𝜔 = 𝝫₀ ≤ 𝝫₁≤ 𝝫₂≤ … ≤𝝫𝜔<𝞲 Where 𝝫𝜔 = ⌵ [sub]i≥0𝝫i. Finally we obtain a characterisation of the finite simple Ockham algebras.
Thesis, PhD Doctor of Philosophy
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