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Subalgebra lattices of a partial unary algebra

Pióro K.

Demonstratio Mathematica

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2012

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Vol. 45, nr 4

755-777

EN

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.

2

The Trawne Member (Albian-Cenomanian, Pieniny Klippen Belt, Carpathians) : a new insight into its foraminiferal assemblages and biostratigraphy

Pióro K.

Studia Geologica Polonica

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2005

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Vol. 124

237-248

EN

Detailed micropalaeontological analysis of samples from the Pasieczny and Trawne stream sections has been undertaken. Twenty five species including seventeen planktonic taxa have been identified. The Rotalipora ticinensis - Planomalina praebuxtorfi new biozone has been proposed. In the Pasieczny Stream section, turbiditic sedimentation commenced during the Early Cenomanian (Rotalipora appenninica Zone). The new biozone R. greenhornensis has been established. The Rotalipora reicheli - Rotalipora greenhornensis Zone, based on coexistence of both nominal species was recognized. The studied foraminiferal associations have confirmed palaeobathymetrical associa- tions B1-B2 (middle part of the continental slope).

3

A few notes on subalgebra lattices, Part II

Pióro K.

Demonstratio Mathematica

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2001

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Vol. 34, nr 1

32-42

EN

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.

4

A few notes on subalgebra lattices, part 1

Pióro K.

Demonstratio Mathematica

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2000

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Vol. 33, nr 4

695-706

EN

First, we apply results proved in [Pió1] and some results of graph theory to formulate and prove a necessary condition for partial (and thus also total) unary algebras to have isomorphic (strong) subalgebra lattices. Although this condition is not sufficient for arbitrary partial unary algebras, we can form, having this fact, a lot of new partial unary algebras with the same subalgebra lattices. Moreover, we use this result to characterize arbitrary two partial (thus in particular also total) monounary algebras with isomorphic (strong) subalgebra lattices. Having this result we can also describe all pairs (A, L), where A is a partial monounary algebra and L a lattice, such that the subalgebra lattice of A is isomorphic to L. In the next part [Pió2] we apply the results of this paper to characterize connections between weak and strong subalgebra lattices of partial (thus also total) monounary algebras.