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IRREDUCIBLE QUASIORDERS OF MONOUNARY ALGEBRAS

Part of: Graph theory Ordered structures Lattices

Published online by Cambridge University Press:  01 March 2013

DANICA JAKUBÍKOVÁ-STUDENOVSKÁ ,
REINHARD PÖSCHEL  and
SÁNDOR RADELECZKI
DANICA JAKUBÍKOVÁ-STUDENOVSKÁ
Affiliation:
Institute of Mathematics, P.J. Šafárik University, Košice, Slovakia (email: [email protected])
REINHARD PÖSCHEL*
Affiliation:
Institute of Algebra, TU Dresden, Germany (email: [email protected])
SÁNDOR RADELECZKI
Affiliation:
Institute of Mathematics, University of Miskolc, Hungary (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Rooted monounary algebras can be considered as an algebraic counterpart of directed rooted trees. We work towards a characterization of the lattice of compatible quasiorders by describing its join- and meet-irreducible elements. We introduce the limit $\cB _\infty $ of all $d$-dimensional Boolean cubes $\Two ^d$ as a monounary algebra; then the natural order on $\Two ^d$ is meet-irreducible. Our main result is that any completely meet-irreducible quasiorder of a rooted algebra is a homomorphic preimage of the natural partial order (or its inverse) of a suitable subalgebra of $\cB _\infty $. For a partial order, it is known that complete meet-irreducibility means that the corresponding partially ordered structure is subdirectly irreducible. For a rooted monounary algebra it is shown that this property implies that the unary operation has finitely many nontrivial kernel classes and its graph is a binary tree.

Keywords

partial ordermonounary algebraquasiorder latticemeet-irreducible quasiordercritical pairrooted tree

MSC classification

Secondary: 06F99: None of the above, but in this section 06B99: None of the above, but in this section 05C05: Trees
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 3 , December 2012 , pp. 259 - 276
Copyright
Copyright © 2013 Australian Mathematical Publishing Association Inc. 

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