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Subdirect products of totally ordered BCK-algebras. (English) Zbl 0632.06017
Let \(L=(L;.,0)\) be an algebra of type (2,0). On L it is possible to introduce a binary relation \(\leq\) as follows: \(x\leq y\) if and only if \(xy=0\). The algebra L is said to be a BCK-algebra if the identities \(x0=x\) and (xy)(xz)\(\leq zy\) are satisfied and \(\leq\) is a partial order with the smallest element 0.
The author characterizes the class of all BCK-algebras which are subdirect products of (totally) ordered BCK-algebras. In more details, he proves that a BCK-algebra is from this class if and only if xy\(\wedge yx\) exists for any x, y and \(xy\wedge yx=0\). This is a generalization of a similar result obtained for commutative BCK-algebras by W. H. Cornish, T. Sturm and T. Traczyk [Math. Jap. 29, 309-320 (1984; Zbl 0579.03049)].
Reviewer: T.Katriňák

06D99 Distributive lattices
06F05 Ordered semigroups and monoids
03G25 Other algebras related to logic
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