We describe -sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine -sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence -permutable -sets for .
In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal of a partial monounary algebra we define the quotient partial monounary algebra . Let , be partial monounary algebras. We describe all partial monounary algebras such that is an ideal of and is isomorphic to .
Mono-unary algebras may be used to construct homomorphisms, subalgebras, and direct products of algebras of an arbitrary type.
In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras and , their weak subalgebra lattices are isomorphic if and only...
One of the main aims of the present and the next part [15] is to show that the theory of graphs (its language and results) can be very useful in algebraic investigations. We characterize, in terms of isomorphisms of some digraphs, all pairs , where is a finite unary algebra and a finite lattice such that the subalgebra lattice of is isomorphic to . Moreover, we find necessary and sufficient conditions for two arbitrary finite unary algebras to have isomorphic subalgebra lattices. We solve...
We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to and , respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary and sufficient conditions are found for quadruples of lattices for which there is a finite unary algebra having...
This paper is a continuation of [5], where -homogeneous and -set-homogeneous algebras were defined. The definitions are analogous to those introduced by Fraïssé [2] and Droste, Giraudet, Macpherson, Sauer [1] for relational structures. In [5] we found all 2-homogeneous and all 2-set-homogeneous monounary algebras when the homogenity is considered with respect to subalgebras, to connected subalgebras and with respect to connected partial subalgebras, respectively. The results of [3], where all...