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On some congruences of power algebras

Agata Pilitowska, Anna Zamojska-Dzienio (2012)

Open Mathematics

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...

On some constructions of algebraic objects

Miroslav Novotný (2006)

Czechoslovak Mathematical Journal

Mono-unary algebras may be used to construct homomorphisms, subalgebras, and direct products of algebras of an arbitrary type.

On some finite groupoids with distributive subgroupoid lattices

Konrad Pióro (2002)

Discussiones Mathematicae - General Algebra and Applications

The aim of the paper is to show that if S(G) is distributive, and also G satisfies some additional condition, then the union of any two subgroupoids of G is also a subgroupoid (intuitively, G has to be in some sense a unary algebra).

On some non-obvious connections between graphs and unary partial algebras

Konrad Pióro (2000)

Czechoslovak Mathematical Journal

In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras 𝐀 and 𝐁 , their weak subalgebra lattices are isomorphic if and only...

On some ternary operations in knot theory

Maciej Niebrzydowski (2014)

Fundamenta Mathematicae

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.

On strong regularity of relations

Josef Šlapal (1994)

Mathematica Bohemica

There exists a natural extension of the notion of preorder from binary relations onto relations whose arities are arbitrary ordinals. In the article we find a condition under which extended preorders coincide with preorders if viewed categorically.

On subalgebra lattices of a finite unary algebra. I.

Konrad Pióro (2001)

Mathematica Bohemica

One of the main aims of the present and the next part [15] is to show that the theory of graphs (its language and results) can be very useful in algebraic investigations. We characterize, in terms of isomorphisms of some digraphs, all pairs 𝐀 , 𝐋 , where 𝐀 is a finite unary algebra and L a finite lattice such that the subalgebra lattice of 𝐀 is isomorphic to 𝐋 . Moreover, we find necessary and sufficient conditions for two arbitrary finite unary algebras to have isomorphic subalgebra lattices. We solve...

On subalgebra lattices of a finite unary algebra. II.

Konrad Pióro (2001)

Mathematica Bohemica

We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs 𝐋 1 , 𝐋 2 of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to 𝐋 1 and 𝐋 2 , respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary and sufficient conditions are found for quadruples 𝐋 1 , 𝐋 2 , 𝐋 3 , 𝐋 4 of lattices for which there is a finite unary algebra having...

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