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On some finite groupoids with distributive subgroupoid lattices

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Pióro K.
Discussiones Mathematicae - General Algebra and Applications
|
2002
|
tom 22
|
nr 1
25-31
EN
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).
2
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The subalgebra lattice of a finite algebra

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Pióro K.
Open Mathematics
|
2014
|
tom 12
|
nr 7
1052-1108
EN
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 general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.
3
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n-Functional digraphs uniquely determined by the skeleton

100%
Pióro K.
Colloquium Mathematicum
|
2002
|
tom 91
|
nr 1
79-89
EN
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.
4
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n-functionality of graphs

100%
Pióro K.
Colloquium Mathematicum
|
2001
|
tom 90
|
nr 2
269-275
EN
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.
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