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MATHEMATICA BOHEMICA, Vol. 126, No. 1, pp. 171-181 (2001)
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# On subalgebra lattices of a finite unary algebra II

## Konrad Pioro

* Konrad Pioro*, Institute of Mathematics, Warsaw University, ul. Banacha 2, PL-02-097 Warsaw, Poland, e-mail: ` kpioro@mimuw.edu.pl`

**Abstract:**
We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs $\langle \mathbf {L}_{1},\mathbf {L}_{2}\rangle$ of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to $\mathbf {L}_{1}$ and $\mathbf {L}_{2}$, 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 $\langle \mathbf {L}_{1},\mathbf {L}_{2}, \mathbf {L}_{3},\mathbf {L}_{4}\rangle$ of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to $\mathbf {L}_{1},\mathbf {L}_{2}, \mathbf {L}_{3},\mathbf {L}_{4}$, respectively.

**Keywords:** graph, finite unary algebra, partial algebra, subalgebras, subalgebra lattices

**Classification (MSC2000):** 05C20, 05C40, 05C99, 08A30, 08A55, 08A60, 05C90, 06B15, 06D05

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