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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

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