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1967•Canadian Journal of MathematicsOpen access

Permutability of Semilattice Congruences on Lattices

Tsuyoshi Fujiwara

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Abstract

Many authors have studied lattice congruences on lattices, but it seems that there are few studies concerning semilattice congruences on lattices. However, it seems that the semilattice congruences on lattices are closely connected with their structure. In this paper, we shall study the characterizations of modular, distributive, and relatively complemented lattices by the permutability of semilattice congruences. We can obtain the dual statements of the following discussion, but we shall not write them as a rule to avoid double descriptions.

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What this paper is about

Many authors have studied lattice congruences on lattices, but it seems that there are few studies concerning semilattice congruences on lattices. However, it seems that the semilattice congruences on lattices are closely connected with their structure. In this paper, we shall study the characterizations of modular, distributive, and relatively complemented lattices by the permutability of semilattice congruences. We can obtain the dual statements of the following discussion, but we shall not write them as a rule to avoid double descriptions.

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Available abstract

Many authors have studied lattice congruences on lattices, but it seems that there are few studies concerning semilattice congruences on lattices. However, it seems that the semilattice congruences on lattices are closely connected with their structure. In this paper, we shall study the characterizations of modular, distributive, and relatively complemented lattices by the permutability of semilattice congruences. We can obtain the dual statements of the following discussion, but we shall not write them as a rule to avoid double descriptions.

Key concepts: Congruence relation, Semilattice, Distributive property, Mathematics, Lattice (music), Pure mathematics, Complete lattice, Distributive lattice

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