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On congruences of weak lattices

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We characterize when an equivalence relation on the base set of a weak lattice \(\mathbf{L}=(L,\sqcup ,\sqcap )\) becomes a congruence on \(\mathbf{L}\) provided it has convex classes. We show that an equivalence relation on L is a congruence on \(\mathbf{L}\) if it satisfies the substitution property for comparable elements. Conditions under which congruence classes are convex are studied. If one fundamental operation of \(\mathbf{L}\) is commutative then \(\mathbf{L}\) is congruence distributive and all congruences of \(\mathbf{L}\) have convex classes.

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Correspondence to Helmut Länger.

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The authors declare that they have no conflict of interests.

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Communicated by A. Di Nola.

Support of the research of both authors by the Austrian Science Fund (FWF), Project I 1923-N25, and the Czech Science Foundation (GAČR), Project 15-34697L, is gratefully acknowledged.

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Chajda, I., Länger, H. On congruences of weak lattices. Soft Comput 20, 4767–4771 (2016).

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