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Subalgebra lattices of a partial unary algebra

Pióro K.

Demonstratio Mathematica

2012

Vol. 45, nr 4

755-777

Necessary and sufficient conditions will be found for quadruples of lattices to be isomorphic to lattices of weak, relative, strong subalgebras and initial segments, respectively, of one partial unary algebra. To this purpose we will start with a characterization of pairs of lattices that are weak and strong subalgebra lattices of one partial unary algebra, respectively. Next, we will describe the initial segment lattice of a partial unary algebra. Applying this result, pairs of lattices of strong subalgebras and initial segments will be characterized. Further, we will characterize pairs of lattices of relative and strong subalgebras and also other pairs of subalgebra lattices of one partial unary algebra.

A few notes on subalgebra lattices, Part II

Pióro K.

Demonstratio Mathematica

2001

Vol. 34, nr 1

32-42

First, we give another, simpler and shorter, proof of the weak subalgebra lattice characterization theorem formulated in [Bar]. Secondly, we apply this result and also facts from [Pió1] to characterize the weak subalgebra lattice of a partial monounary algebra. Using this characterization and results from the previous part [Pió2] we describe all pairs of lattices (L1, L2) for which there is a partial monounary algebra having its weak and strong subalgebra lattices isomorphic to L1 and L2, respectively.