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ON CONNECTIONS BETWEEN HYPERGRAPHS AND ALGEBRAS

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*Konrad Pioro*

**Address.** Institute of Mathematics, Warsaw University, ul. Banacha
2, 02-097 Warsaw, POLAND
**E-mail:** kpioro@mimuw.edu.pl

**Abstract.** 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 \cite{[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 that two partial algebras have isomorphic weak subalgebra lattices
iff their hypergraphs are isomorphic. Thirdly, for an arbitrary lattice
$\mathbf{L}$ and a partial algebra $\mathbf{A}$ we describe (necessary
and sufficient conditions) when the weak subalgebra lattice of $\mathbf{A}$
is isomorphic to $\mathbf{L}$.

**AMSclassification.** 05C65, 05C99, 08A30, 08A55 (05C90, 06B15,
06D05)

**Keywords.** Hypergraph, subalgebras (relative, strong, weak), subalgebra
lattices, partial algebra