On a method of construction of abstract algebras

Płonka, J.

doi:10.4064/fm-61-2-183-189

Cited by 96 publications

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“…The operation * is also a partition operation for (S, •, +) in order to obtain this bisemilattice as a Plonka sum of its maximal distributive sublattice (cf. [14], [15], [22]). In the sequel we call (S, •,+) in this context the Plonka sum of the distributive lattices (S t , -, +) over the semilattice (/, •) by the homomorphisms (P it j)jj and denote the elements of S t by a f , b { , etc.…”

Section: Proposition If (S • +) Is a Distributive Bisemilattice Tmentioning

confidence: 99%

Duality for distributive bisemilattices

Gierz

Romanowska

1991

J Aust Math Soc A

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Section: Proposition If (S • +) Is a Distributive Bisemilattice Tmentioning

confidence: 99%

Duality for distributive bisemilattices

Gierz

Romanowska

1991

J Aust Math Soc A

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“…Equivalently, V = V ∨ Sl ρ , following from the fact that Sl ρ is the class of algebras satisfying all regular identities of type ρ. If V is a strongly irregular variety, there is a very good structure theory for the regularization V (due to P lonka [40,41]), which we shall now describe.…”

Section: P Lonka Sumsmentioning

confidence: 99%

Commutative idempotent groupoids and the constraint satisfaction problem

Bergman

Failing

2015

Algebra Univers.

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A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI groupoid) will be tractable. It is known that every semilattice (i.e., an associative CI groupoid) is tractable. A groupoid identity is of Bol-Moufang type if the same three variables appear on either side, one of the variables is repeated, the remaining two variables appear once, and the variables appear in the same order on either side (for example, x(x(yz)) ≈ (x(xy))z). These identities can be thought of as generalizations of associativity. We show that there are exactly 8 varieties of CI groupoids defined by a single additional identity of Bol-Moufang type, derive some of their important structural properties, and use that structure theory to show that 7 of the varieties are tractable. We also characterize the finite members of the variety of CI groupoids satisfying the self-distributive law x(yz) ≈ (xy)(xz), and show that they are tractable.

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“…The binary operation f : UˆU Þ Ñ U is called non-idempotent Plonka function of U if it satisfies the following identities (cf. [35], [37], [39]):…”

Section: Interlaced Weakly Idempotent (Pre-)bilatticesmentioning

confidence: 99%

“…An algebra U " pU, Σq is called weakly Plonka sum of its subalgebras pU i ; Σq, where i P I, if the following conditions are valid (cf. [35], [37], [39]…”

Section: Interlaced Weakly Idempotent (Pre-)bilatticesmentioning

confidence: 99%

Weakly Idempotent Lattices and Bilattices, Non-Idempotent Plonka Functions

Davidova

Movsisyan

2015

Demonstratio Mathematica

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Abstract. In this paper, we study weakly idempotent lattices with an additional interlaced operation. We characterize interlacity of a weakly idempotent semilattice operation, using the concept of hyperidentity and prove that a weakly idempotent bilattice with an interlaced operation is epimorphic to the superproduct with negation of two equal lattices. In the last part of the paper, we introduce the concepts of a non-idempotent Plonka function and the weakly Plonka sum and extend the main result for algebras with the well known Plonka function to the algebras with the non-idempotent Plonka function. As a consequence, we characterize the hyperidentities of the variety of weakly idempotent lattices, using non-idempotent Plonka functions, weakly Plonka sums and characterization of cardinality of the sets of operations of subdirectly irreducible algebras with hyperidentities of the variety of weakly idempotent lattices. Applications of weakly idempotent bilattices in multi-valued logic is to appear. IntroductionThere exist various extensions of the concept of a lattice. For example, in [14], [13], weakly associative lattices were introduced and in [2], [20], [21], [23], the lattices with a third operation were studied. In [24], an algebra with a system of identities was introduced, which we call weakly idempotent lattices (also see [18], [36]).The paper consists of Introduction and four paragraphs.In the second paragraph, we give the definitions of a weakly idempotent semilattice, a weakly idempotent lattice, a weakly idempotent (pre-)bilattice, an interlaced operation, an interlaced weakly idempotent (pre-)bilattice and 2010 Mathematics Subject Classification: 03G10, 06A12, 06B05, 06B10, 06B15, 06B20, 08A05, 08B26, 08C05, 03C85.Key words and phrases: weakly idempotent semilattice, weakly idempotent lattice, weakly idempotent bilattice, interlaced operation, interlaced weakly idempotent bilattice, hyperidentity, non-idempotent Plonka function, weakly Plonka sum, weakly idempotent quasilattice. hyperidentities; then we prove some preliminary results. Further, we establish a connection among these concepts of these weakly idempotent structures and the corresponding quasiorders (Lemmas 2.7-2.10, Corollaries 2.12, 2.13).In the third paragraph, we prove some properties of weakly idempotent lattices. In particular, in Theorem 3.3 we characterize interlacity for the weakly idempotent semilattice operation, using the concept of hyperidentity. In paragraph four, we characterize the interlaced weakly idempotent bilattices (Theorem 4.7) and the weakly idempotent pre-bilattices (Corollary 4.8). As a corollary we also obtain a characterization of weakly idempotent distributive bilattices (Corollary 4.9). In the chapter fifth, we introduce the concepts of a non-idempotent Plonka function and a weakly Plonka sum. Here the main result for algebras with the well known Plonka function is extended to the algebras with a non-idempotent Plonka function. In the last chapter, as a corollary we characterize hyperidentities of the variety ...

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On a method of construction of abstract algebras

Cited by 96 publications

References 0 publications

Duality for distributive bisemilattices

Duality for distributive bisemilattices

Commutative idempotent groupoids and the constraint satisfaction problem

Weakly Idempotent Lattices and Bilattices, Non-Idempotent Plonka Functions

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