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On the number of finite algebraic structures

Erhard Aichinger, Peter Mayr, R. McKenzie (2014)

Journal of the European Mathematical Society

We prove that every clone of operations on a finite set A , if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting R for some finitary relation R over A . It follows that for a fixed finite set A , the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few...

One interval in the lattice of partial hyperclones

Rade Doroslovački, Jovanka Pantović, Gradimir Vojvodić (2005)

Czechoslovak Mathematical Journal

In this paper the structure of the interval [ O A , H p A ] in the lattice of partial hyperclones is determined, where O A is the clone of all total operations and H p A is the clone of all partial hyperoperations on A .

Order affine completeness of lattices with Boolean congruence lattices

Kalle Kaarli, Vladimir Kuchmei (2007)

Czechoslovak Mathematical Journal

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices 𝐋 easily reduces to the case when 𝐋 is a subdirect product of two simple lattices 𝐋 1 and 𝐋 2 . Our main result claims that such a lattice is locally...

Orthorings

Ivan Chajda, Helmut Länger (2004)

Discussiones Mathematicae - General Algebra and Applications

Certain ring-like structures, so-called orthorings, are introduced which are in a natural one-to-one correspondence with lattices with 0 every principal ideal of which is an ortholattice. This correspondence generalizes the well-known bijection between Boolean rings and Boolean algebras. It turns out that orthorings have nice congruence and ideal properties.

Polyabelian loops and Boolean completeness

François Lemieux, Cristopher Moore, Denis Thérien (2000)

Commentationes Mathematicae Universitatis Carolinae

We consider the question of which loops are capable of expressing arbitrary Boolean functions through expressions of constants and variables. We call this property Boolean completeness. It is a generalization of functional completeness, and is intimately connected to the computational complexity of various questions about expressions, circuits, and equations defined over the loop. We say that a loop is polyabelian if it is an iterated affine quasidirect product of Abelian groups; polyabelianness...

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