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Trees as commutative BCK-Algebras

Published online by Cambridge University Press:  17 April 2009

William H. Cornish
William H. Cornish
Affiliation:
School of Mathematical Sciences, Flinders University of South Australia, Bedford Park, South Australia 5042, Australia.
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Abstract

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A new method of constructing commutative BCK-algebras is given. It depends upon the notion of a valuation of a lower semilattice in a given commutative BCK-algebra. Any tree vith the descending chain condition has a valuation in the natural numbers, considered as a commutative BCK-algebra; the valuation is the height-function. Thus, any tree of finite height possesses a uniquely determined commutative BCK-structure. The finite trees with at most one atom and height at most n are precisely the finitely generated subdirectly irreducible (simple) algebras in the subvariety of commutative BCK-algebras which satisfy the identity (En): xyn = xyn+1. Due to congruence-distributivity, it is then possible to describe the associated lattice of subvarieties.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 2 , April 1981 , pp. 181 - 190
Copyright
Copyright © Australian Mathematical Society 1981

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