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Keywords:
graph; finite unary algebra; partial algebra; subalgebras; subalgebra lattices
Summary:
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.
References:
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