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A few notes on subalgebra lattices, Part II

100%

Pióro K.

Demonstratio Mathematica

2001

tom 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.

Normal pseudo-BCK-algebras

80%

Dymek G.

Commentationes Mathematicae

2011

tom Vol. 51, [Z] 1

99-107

A normal pseudo-BCK-algebra X is an algebra in which every subalgebra of X is an ideal of X. Characterizations of normal pseudo-BCK-algebras are given.

A few notes on subalgebra lattices, part 1

80%

Pióro K.

Demonstratio Mathematica

2000

tom Vol. 33, nr 4

695-706

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.

Normal pseudo-BCK-algebras

80%

Dymek G.

Commentationes Mathematicae

2011

tom 51

nr 1

A normal pseudo-BCK-algebra \(X\) is an algebra in which every subalgebra of \(X\) is an ideal of \(X\). Characterizations of normal pseudo-BCK-algebras are given.

A note on semidirect sum of Lie algebras

51%

Ostrowski T.

Discussiones Mathematicae - General Algebra and Applications

2013

tom 33

nr 2

233-247

In the paper there are investigated some properties of Lie algebras, the construction which has a wide range of applications like computer sciences (especially to computer visions), geometry or physics, for example. We concentrate on the semidirect sum of algebras and there are extended some theoretic designs as conditions to be a center, a homomorphism or a derivative. The Killing form of the semidirect sum where the second component is an ideal of the first one is considered as well.