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4 2011

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Embedding sums of cancellative modes into functorial sums

Romanowska A. B., Stronkowski M., Zamojska-Dzienio A.

Demonstratio Mathematica

2011

Vol. 44, nr 3

557-569

The paper discusses a representation of modes (idempotent and entropic algebras)as subalgebras of so called functorial sums of cancellative algebras.We show that each mode that has a homomorphism onto an algebra satisfying a certain additional condition,with corresponding cancellative congruence classes,embeds into a functorial sum of cancellative algebras.We also discuss typical applications of this result.

Embedding modes into semimodules, part III

Pilitowska A., Romanowska A. B.

Demonstratio Mathematica

2011

Vol. 44, nr 4

791-800

In the first part of this paper, we considered the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provided a general construction of such semirings, along with basic examples and some general properties. In the second part of the paper we discussed some selected varieties of modes, in particular, varieties of affine spaces, varieties of barycentric algebras and varieties of semilattice modes, and described the semirings determining their semi-linearizations, the varieties of semimodules having these algebras as idempotent subreducts. The third part is devoted to varieties of differential groupoids and more general differential modes, and provides the semirings of the semi-linearizations of these varieties.

Embedding modes into semimodules, part II

Pilitowska A., Romanowska A. B.

Demonstratio Mathematica

2011

Vol. 44, nr 4

781-790

The first part of this paper specified the semi-affinization semiring of a mode variety as the universal scalar semiring for semimodules whose idempotent subreducts lie in the given variety of modes. The current part of the paper focusses on some selected varieties of modes (affine spaces, barycentric algebras, semilattice modes), and computes the semi-affinization semirings of these varieties.

Embedding modes into semimodules, part I

Pilitowska A., Romanowska A. B.

Demonstratio Mathematica

2011

Vol. 44, nr 3

523-534

By recent results of M. Stronkowski, it is known that not all modes embed as subreducts into semimodules over commutative unital semirings. Related to this problem is the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provide a general construction of such semirings, along with basic examples and some general properties. The second part of the paper will deal with applications of the general construction to some selected varieties of modes, and will provide a description of semirings determining varieties of semimodules having algebras from these varieties as idempotent subreducts.