On some unary algebras and their subalgebra lattices

We first define lattices, called normal, which are uniquely represented by directed graphs. Secondly, we describe all unary algebras (called normal, too) such that their subalgebra lattices are normal. Next, we characterize pairs (A,L) such that the subalgebra lattice of A is isomorphic to L, where A is a normal unary algebra and L is a normal lattice. Further, we describe pairs of normal unary algebras with isomorphic subalgebra lattices. We use these results in the second part of the paper to find necessary and sufficient conditions for pairs of lattices to be isomorphic to a pair of the weak and strong subalgebra lattices of one normal unary algebra

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

summary: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 characterize the weak subalgebra lattice for algebras of some types

On some finite groupoids with distributive subgroupoid lattices

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 a property of neighborhood hypergraphs

summary: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 \Cal G and a hypergraph \Cal H have the same neighborhood hypergraph and the neighborhood relation in \Cal G is a subrelation of such a relation in \Cal H, then \Cal H is inscribed into \Cal G (both seen as coverings). In particular, if \Cal H is also a clique hypergraph, then \Cal H = \Cal G

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

summary: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 characterize the weak subalgebra lattice for algebras of some types