On subalgebra lattices of a finite unary algebra. II. (English) Zbl 0978.08004

Pióro, Konrad

Math. Bohem. 126, No. 1, 171-181 (2001).

Summary: We use graph-algebraic results from the first part of this paper [ibid. 126, No. 1, 161-170 (2001; Zbl 0978.08003), reviewed above] and some results of graph theory to characterize all pairs \(\langle \mathcal L_{1},\mathcal L_{2}\rangle \) of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to \(\mathcal L_{1}\) and \(\mathcal L_{2}\), 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 \(\langle \mathcal L_{1},\mathcal L_{2}, \mathcal L_{3},\mathcal L_{4}\rangle \) of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to \(\mathcal L_{1},\mathcal L_{2}, \mathcal L_{3},\mathcal L_{4}\), respectively.

MSC:

08A60	Unary algebras
08A30	Subalgebras, congruence relations
06B15	Representation theory of lattices
05C20	Directed graphs (digraphs), tournaments
08A55	Partial algebras

Keywords:

graph; finite unary algebra; partial algebra; subalgebras; subalgebra lattices