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IRREDUCIBLE QUASIORDERS OF MONOUNARY ALGEBRAS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 93 / Issue 3 / December 2012
- Published online by Cambridge University Press:
- 01 March 2013, pp. 259-276
- Print publication:
- December 2012
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- Article
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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.
