Differential Modes

Kravchenko, A. V.; Pilitowska, Agata; Romanowska, Anna; Stanovský, David

doi:10.1142/s0218196708004561

Cited by 10 publications

(16 citation statements)

References 9 publications

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“…However, as noted in [3], it would be very easy to extend all the notions and results of these papers to modes with one n-ary operation for all n ≥ 4. The ternary case was chosen only to avoid technical complications.…”

Section: The Semiring Of the Variety Of Differential Modesmentioning

confidence: 99%

“…Ternary counterparts of differential groupoids, so-called (ternary) differential modes, are discussed in [3] and [6]. However, as noted in [3], it would be very easy to extend all the notions and results of these papers to modes with one n-ary operation for all n ≥ 4.…”

Section: The Semiring Of the Variety Of Differential Modesmentioning

confidence: 99%

“…Apart from having interesting properties of their own, differential groupoids and differential modes play an important role in the problem of embedding modes into semimodules, and also in the theory of finitely generated modes. (See [2], [3], [6], [9], [10]. ) We use the terminology, notation and results of the first and second parts of the paper, and continue the section numbering from the second paper.…”

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Embedding modes into semimodules, Part III

Pilitowska

Romanowska

2011

Demonstratio Mathematica

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Abstract. 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. This paper is a direct continuation of the first and second parts appearing with the same title [4] and [5]. In the first part, 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, and such that each semimodule-embeddable member of this mode variety embeds into a semimodule over the constructed semiring. We described a general construction of such semirings, with basic examples and some general properties. In the second part, we investigated selected varieties of modes, and described the semirings determining varieties of semimodules having algebras of these classes as subreducts, and discussed properties of the corresponding semi-affine spaces. In particular, we investigated varieties of affine spaces, varieties of barycentric algebras and varieties of semilattice modes. The third part is devoted to varieties of differential groupoids and more general differ-2000 Mathematics Subject Classification: 08A62, 08A05, 08B, 16Y60, 20N02, 20N10. Key words and phrases: modes (idempotent entropic algebras), idempotent reducts and subreducts of (semi)modules over commutative (semi)rings, differential groupoids, differential modes.

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Section: The Semiring Of the Variety Of Differential Modesmentioning

confidence: 99%

Section: The Semiring Of the Variety Of Differential Modesmentioning

confidence: 99%

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Embedding modes into semimodules, Part III

Pilitowska

Romanowska

2011

Demonstratio Mathematica

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“…The study of general differential modes was initiated in [5], which explains our motivation and contains the following results: alternative axiomatizations, a description of reducts, a decomposition based on the congruence λ (to be defined in Section 2), a description of free algebras, including normal forms of terms, a proof that finitely based subvarieties are relatively 1-based, and an example of a locally finite subvariety with no finite base for its identities. The second paper of the series [7] contains a thorough discussion of the lattice of subvarieties of Szendrei differential modes, including the fact that all of them are finitely based (actually in two variables).…”

Section: Introductionmentioning

confidence: 99%

“…The binary case is significantly simpler, because binary modes are Szendrei modes (see Section 4 for a discussion). The results of [9,10] and some of [12] were appropriately generalized in [5,7], however our previous two papers did not address subdirectly irreducible algebras. The present paper fills the gap.…”

Section: Introductionmentioning

confidence: 99%

Subdirectly Irreducible Differential Modes

Stanovský

2012

Int. J. Algebra Comput.

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Abstract. We study a variety of modes (idempotent algebras with mutually commuting term operations), so called differential modes, having a strongly solvable chain 0 ≤ α ≤ 1 in their congruence lattices. We show an explicit description of subdirectly irreducible algebras in this variety, and use it to compute residual bounds of its subvarieties. It follows from our results that all subvarieties with a finite residual bound are finitely based.

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