Types of polynomial completeness of expanded groups

Aichinger, Erhard; Mudrinski, Nebojša

doi:10.1007/s00012-009-2136-y

Cited by 9 publications

(7 citation statements)

References 11 publications

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“…Let be a group, written additively but not necessarily abelian, with neutral element 0. The structure of the near-ring of zero fixing congruence preserving functions on G has been the topic of several previous investigations [ 1 , 4 ]. In this paper we initiate the study of characterizing those groups G such that is a ring.…”

Section: Introduction: Background and Notationmentioning

confidence: 99%

Rings of congruence preserving functions

Maxson

Saxinger

2017

Monatsh Math

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Section: Introduction: Background and Notationmentioning

confidence: 99%

Rings of congruence preserving functions

Maxson

Saxinger

2017

Monatsh Math

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“…Proof: Since every weakly 1-polynomially rich algebra is 1-affine complete algebra the statment can be obtained from Proposition 9.5 in [8].…”

Section: Then We Havementioning

confidence: 99%

O polinomima u algebrama Maljceva

Mudrinski¹

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“…In this section we introduce the notion of homogeneous congruence (cf. [18,5,6]), and we describe the properties of homogeneous congruences in Mal'cev algebras with (SC1). Following [6] we define: Definition 6.1 ([6, Definition 8.1]).…”

Section: Homogeneous Congruences In Mal'cev Algebras With (Sc1)mentioning

confidence: 99%

“…[18,5,6]), and we describe the properties of homogeneous congruences in Mal'cev algebras with (SC1). Following [6] we define: Definition 6.1 ([6, Definition 8.1]). A Mal'cev algebra A satisfies the condition (APMI) if for all strictly meet irreducible congruences α, β of A such that…”

Section: Homogeneous Congruences In Mal'cev Algebras With (Sc1)mentioning

confidence: 99%

“…If each k-ary operation on A that preserves the congruences of A is a polynomial function of A, then A is called k-affine complete. In [6], E. Aichinger and N. Mudrinski proved a characterization of k-affine complete finite algebras with a group reduct (also known in the literature as expanded groups) whose lattice of ideals satisfies a property called (APMI).…”

Section: Introductionmentioning

confidence: 99%

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Polynomial completeness criteria in finite Boolean algebras

Romov

Proceedings of 26th IEEE International Symposium on Multiple-Valued Logic (ISMVL'96)

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For a given finite Boolean algebra with r (r 2 2) atoms we consider the set BF(r) of all polynomials produced bysuperpositions of the main operations and r atomic constants. Using the isomoFhism between BF(r) and P; , the arity-calibrated product of r two-valued logic algebras Pz , and also the description of all maximal subalgebras of P i , we establish a general completeness criterion in BF(r), a Sheer criterion for a single Boolean function to be a generating element in BF(r), and Slupecki type criterion in BF(r) as well.We consider finite Boolean algebras as finite distributive lattices with complement. Due to a well-known result ( see e.g.[l],Ch.V) any algebra of such kind is isomorphic to the 2' -element Boolean e e b r a on the set of all subsets of r-element set of its atoms. So, in what follows, Rr) is the set of all subsets of r = {e,, ..., e, } ( r 2 2 ) and C wr); 0 , el,.*., e,, U, n, 1 > is the Boolean algebra with r atoms having in its signature the empty set, the atoms, the union, the intersection and the complement.The polynomials in this algebra (i.e. operations constructed from the main operations, the projections, and the atomic constants by superposition) are called Boolean functions or Boolean operations of rank r. Denote by BF(r) the set of all Boolean functions. Functions from BF(r), r 2 2 , play an essential role as building blocks of set-valued logic networks which serve as a mathematical apparatus for the wave-parallel (wire-free) computing architecture design (see e.g. [2]).Example. Let r=2. Hence, we have 4-element set B (2) with atoms e, and e,. Assume #l(X)=(ie, n X)U(e, n l x ) , MN= ( l e , n x)u(~, n W, then hm,, x,, 262 0-8186-7392-3/96 $05.00 0 1996 IEEE

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Types of polynomial completeness of expanded groups

Cited by 9 publications

References 11 publications

Rings of congruence preserving functions

Rings of congruence preserving functions

O polinomima u algebrama Maljceva

Polynomial completeness criteria in finite Boolean algebras

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