An Invitation to General Algebra and Universal Constructions

Bergman, George M.

doi:10.1007/978-3-319-11478-1

Cited by 88 publications

(67 citation statements)

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“…a^pb _ aq " a^a, a _ pb^aq " a _ a, (weakly absorption) (4) a^a " a _ a. (equalization) (5) The operation _ is called sum.…”

Section: Preliminary Concepts and Resultsmentioning

confidence: 99%

“…Hyperidentities are usually written without quantifiers, w 1 " w 2 . We say that the hyperidentity w 1 " w 2 is satisfied in the algebra pQ; F q if this equality is valid, when every object variable and every functional variable in it is replaced by any element from Q and by any operation of the corresponding arity from F (supposing the possibility of such replacement) ( [27]- [29], [43], [4]). …”

Section: Preliminary Concepts and Resultsmentioning

confidence: 99%

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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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“…a^pb _ aq " a^a, a _ pb^aq " a _ a, (weakly absorption) (4) a^a " a _ a. (equalization) (5) The operation _ is called sum.…”

Section: Preliminary Concepts and Resultsmentioning

confidence: 99%

Section: Preliminary Concepts and Resultsmentioning

confidence: 99%

Weakly Idempotent Lattices and Bilattices, Non-Idempotent Plonka Functions

Davidova

Movsisyan

2015

Demonstratio Mathematica

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“…Since the proofs are very similar, we only give a proof of (2). Suppose that f → is an order embedding in and f u ≤ f v for some u v → .…”

Section: Monomorphismsmentioning

confidence: 97%

“…We start with monomorphisms and submonomorphisms. (1) monomorphisms in the category are precisely the injective homomorphisms; (2) submonomorphisms in the category are precisely the order embeddings.…”

Section: Monomorphismsmentioning

confidence: 99%