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The subalgebra lattice of a finite algebra

Konrad Pióro — 2014

Open Mathematics

The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more...

n-functionality of graphs

Konrad Pióro — 2001

Colloquium Mathematicae

We first characterize in a simple combinatorial way all finite graphs whose edges can be directed to form an n-functional digraph, for a fixed positive integer n. Next, we prove that the possibility of directing the edges of an infinite graph to form an n-functional digraph depends on its finite subgraphs only. These results generalize Ore's result for functional digraphs.

n-Functional digraphs uniquely determined by the skeleton

Konrad Pióro — 2002

Colloquium Mathematicae

We show that any total n-functional digraph D is uniquely determined by its skeleton up to the orientation of some cycles and infinite chains. Next, we characterize all graphs G such that each n-functional digraph obtained from G by directing all its edges is total. Finally, we describe finite graphs whose edges can be directed to form a total n-functional digraph without cycles.

On a property of neighborhood hypergraphs

Konrad Pióro — 2006

Commentationes Mathematicae Universitatis Carolinae

The aim of the paper is to show that no simple graph has a proper subgraph with the same neighborhood hypergraph. As a simple consequence of this result we infer that if a clique hypergraph 𝒢 and a hypergraph have the same neighborhood hypergraph and the neighborhood relation in 𝒢 is a subrelation of such a relation in , then is inscribed into 𝒢 (both seen as coverings). In particular, if is also a clique hypergraph, then = 𝒢 .

Some properties of the weak subalgebra lattice of a partial algebra of a fixed type

Konrad Pióro — 2002

Archivum Mathematicum

We investigate, using results from [[p3]], when a given lattice is isomorphic to the weak subalgebra lattice of a partial algebra of a fixed type. First, we reduce this problem to the question when hyperedges of a hypergraph can be directed to a form of directed hypergraph of a fixed type. Secondly, we show that it is enough to consider some special hypergraphs. Finally, translating these results onto the lattice language, we obtain necessary conditions for our algebraic problem, and also, we completely...

On connections between hypergraphs and algebras

Konrad Pióro — 2000

Archivum Mathematicum

The aim of the present paper is to translate some algebraic concepts to hypergraphs. Thus we obtain a new language, very useful in the investigation of subalgebra lattices of partial, and also total, algebras. In this paper we solve three such problems on subalgebra lattices, other will be solved in [[Pio4]]. First, we show that for two arbitrary partial algebras, if their directed hypergraphs are isomorphic, then their weak, relative and strong subalgebra lattices are isomorphic. Secondly, we prove...

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 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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