On some graph problem in the theory of partial algebras—Part I Original Research Article

In the theory of partial algebras we have, in a natural way, the following algebraic problem: When is a unary partial algebra A of type K uniquely determined (up to isomorphism) by its weak subalgebra lattice in the class of all unary partial algebras of type K? Such monounary partial algebras are characterized in Bartol (Comment. Math. Univ. Carolin. 31 (1990) 411). Unfortunately, from Bartol (Comment. Math. Univ. Carolin. 31 (1990) 405) and Pióro (Czechoslovak Math. J. 50 (125) (2000) 295) it follows that for other classes of unary algebras such a full reconstruction, without additional restrictions onto classes, is rather impossible. On the other hand, each unary partial algebra A of type K can be represented by a digraph D(A) obtained from A by omitting names of operations, and by a graph D∗(A) obtained from the first by omitting the orientation of all edges (see Bartol, 1990 and Pióro, 2000). Secondly, Bartol (1990) shows that D∗(A) uniquely determines the weak subalgebra lattice of A. Thirdly, by Pióro (2000), if A is a unary algebra of type K, then D(A) is of type |K|, i.e. at most |K| edges start from each vertex. Thus we obtain the following graph problem, which will be solved in this and the next part (On some graph problem in the theory of partial algebras, part II): When is a digraph D of type η (where η is a cardinal number) uniquely determined by its graph D∗ in the class of all digraphs of the same type η?