### A note on tolerance lattices of algebras with restricted similarity type

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We introduce a special set of relations called clausal relations. We study a Galois connection Pol-CInv between the set of all finitary operations on a finite set D and the set of clausal relations, which is a restricted version of the Galois connection Pol-Inv. We define C-clones as the Galois closed sets of operations with respect to Pol-CInv and describe the lattice of all C-clones for the Boolean case D = {0,1}. Finally we prove certain results about C-clones over a larger set.

Four-part semigroups form a new class of semigroups which became important when sets of Boolean operations which are closed under the binary superposition operation f + g := f(g,...,g), were studied. In this paper we describe the lattice of all subsemigroups of an arbitrary four-part semigroup, determine regular and idempotent elements, regular and idempotent subsemigroups, homomorphic images, Green's relations, and prove a representation theorem for four-part semigroups.

A construction of all homomorphisms of an algebra with a finite number of operations into an algebra of the same type is presented that consists in replacing algebras by suitable mono-unary algebras (possibly with some nullary operations) and their homomorphisms by suitable homomorphisms of the corresponding mono-unary algebras. Since a construction of all homomorphisms between two mono-unary algebras is known (see, e.g., [6], [7], [8]), a construction of all homomorphisms of an arbitrary algebra...

A term operation implication is introduced in a given basic algebra $\mathcal{A}$ and properties of the implication reduct of $\mathcal{A}$ are treated. We characterize such implication basic algebras and get congruence properties of the variety of these algebras. A term operation equivalence is introduced later and properties of this operation are described. It is shown how this operation is related with the induced partial order of $\mathcal{A}$ and, if this partial order is linear, the algebra $\mathcal{A}$ can be reconstructed by means of...

We find the basis of all linear identities which are true in the variety of entropic graph algebras. We apply it to describe the lattice of all subvarieties of power entropic graph algebras.

A triple-semilattice is an algebra with three binary operations, which is a semilattice in respect of each of them. A trice is a triple-semilattice, satisfying so called roundabout absorption laws. In this paper we investigate distributive trices. We prove that the only subdirectly irreducible distributive trices are the trivial one and a two element one. We also discuss finitely generated free distributive trices and prove that a free distributive trice with two generators has 18 elements.

In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class...

We introduce a way to color the regions of a classical knot diagram using ternary operations, so that the number of colorings is a knot invariant. By choosing appropriate substitutions in the algebras that we assign to diagrams, we obtain the relations from the knot group, and from the core group. Using the ternary operator approach, we generalize the Dehn presentation of the knot group to extra loops, and a similar presentation for the core group to the variety of Moufang loops.