The is a temporarty set of notes based on several older (and some ancient) notes. In due course they will be re-done. Much of this is extracted from [5], but I don’t recommend reading that. Let Λ be an arbitrary lattice. We always assume the Λ has a top > and a bottom ⊥. Later we need to assume that Λ is modular, and perhaps even later that it is an idiom, but not just yet. Let Cong(Λ) be the f...

We describe G-sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine G-sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence n-permutable G-sets for n = 2, 2.5, 3.

A universal algebra is simple if and only if it has more than one element and there are no proper homomorphisms defined upon it. In this note we study a related notion : an algebra is pseudo-simple if and only if it has more than one element and is isomorphic to all its proper (i.e., non-one-element) homomorphic images. The two concepts simplicity and pseudo-simplicity are examples of concepts ...

In this work we introduce the notion of fuzzy congruence relation on an ndgroupoid and study conditions on the nd-groupoid which guarantee a complete lattice structure on the set of fuzzy congruence relations. The study of these conditions allowed to construct a counterexample to the statement that the set of fuzzy congruences on a hypergroupoid is a complete lattice.

Let L be a congruence-uniform lattice. In this note, we investigate the shard order on L that was introduced by N. Reading. When L is a poset of regions of a hyperplane arrangement the shard order always is a lattice. For general L, however, this fails. We provide a necessary condition for the shard order to be a lattice, and we show how to construct a congruence-uniform lattice L′ from L such ...

In this paper, the notions of fuzzy soft subalgebra and fuzzy soft convex subalgebra of a residuated lattice are introduced and some related properties are investigated. Then, we define fuzzy soft congruence on a residuated lattice and obtain the relation between fuzzy soft convex subalgebras and fuzzy soft congruence relations on residuated lattices. The concept of soft homomorphism is defined...

The number of distinct operations (that is, the range of the variable a) may be infinite, but for our main result (Theorem 2), we shall require every n(a) to be finite—that is, it will concern algebras with finitary operations. The concepts of subalgebra, congruence relation on an algebra, homomorphism of one algebra A onto (or into) another algebra with the same operations, and of the direct u...

Hotzel asks this for semigroups and right congruences; his and the above version are equivalent. Left congruences on M are equivalence relations closed under all left translations; these are thus equivalent to congruences on the free left M -set on one generator. When M is a group, any left congruence is the decomposition of M into the left cosets of some subgroup, so the lattice of left congru...