An M-Set is a unary algebra $\langle X,M\rangle$ whose set $M$ of operations is a monoid of transformations of $X$; $\langle X,M\rangle$ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $\langle X,M\rangle$ if the congruence lattice of $\langle X,M\rangle$ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\Pi \left(L\right)$ is introduced here. $\Pi \left(L\right)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\Pi \left(L\right)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi$-product lattice. A $\Pi$-product...

We investigate the interval $I\left(p^3\right)$ in the lattice of clones on the ring ${\mathbb{Z}}_{p^3}$ between the clone of polynomial operations and the clone of congruence preserving operations. All clones in this interval are known and described by means of generators. In this paper, we characterize each of these clones by the property of preserving a small set of relations. These relations turn out to be in a close connection to commutators.

We give Mal’cev conditions for varieties 4V4 whose congruences on the product $A\times B,A,B\in V$, are determined by their restrictions on the axes in $A\times B$.

Using congruence schemes we formulate new characterizations of congruence distributive, arithmetical and majority algebras. We prove new properties of the tolerance lattice and of the lattice of compatible reflexive relations of a majority algebra and generalize earlier results of H.-J. Bandelt, G. Cz'{e}dli and the present authors. Algebras whose congruence lattices satisfy certain 0-conditions are also studied.

We present a countable infinite chain of conditions which are essentially weaker then congruence modularity (with exception of first two). For varieties of algebras, the third of these conditions, the so called 4-submodularity, is equivalent to congruence modularity. This is not true for single algebras in general. These conditions are characterized by Maltsev type conditions.

The topic of the paper are $\Omega$-algebras, where $\Omega$ is a complete lattice. In this research we deal with congruences and homomorphisms. An $\Omega$-algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an $\Omega$-valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce $\Omega$-valued congruences, corresponding quotient $\Omega$-algebras and $\Omega$-homomorphisms and we investigate connections among these notions. We prove...

Effect basic algebras (which correspond to lattice ordered effect algebras) are studied. Their ideals are characterized (in the language of basic algebras) and one-to-one correspondence between ideals and congruences is shown. Conditions under which the quotients are OMLs or MV-algebras are found.

A ternary ring is an algebraic structure $ℛ=(R;t,0,1)$ of type $(3,0,0)$ satisfying the identities $t(0,x,y)=y=t(x,0,y)$ and $t(1,x,0)=x=(x,1,0)$ where, moreover, for any $a$, $b$, $c\in R$ there exists a unique $d\in R$ with $t(a,b,d)=c$. A congruence $\theta$ on $ℛ$ is called normal if $ℛ/\theta$ is a ternary ring again. We describe basic properties of the lattice of all normal congruences on $ℛ$ and establish connections between ideals (introduced earlier by the third author) and congruence kernels.

In [1] ideals and congruences on semiloops were investigated. The aim of this paper is to generalize results obtained for semiloops to the case of left divisible involutory groupoids.