A Cayley Theorem for Ternary Algebras

Ésik, Zoltán

doi:10.1142/s0218196798000156

Cited by 17 publications

(6 citation statements)

References 4 publications

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“…The binary version of Cayley theorem for Boolean algebras is proved in [3] (also see [4][5][6]). An analogous result is proved for the so-called ternary algebras in [7].…”

Section: Introductionmentioning

confidence: 52%

Binary Representations of Algebras With at Most Two Binary Operations: A Cayley Theorem for Distributive Lattices

Movsisyan

2009

Int. J. Algebra Comput.

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“…The binary version of Cayley theorem for Boolean algebras is proved in [3] (also see [4][5][6]). An analogous result is proved for the so-called ternary algebras in [7].…”

Section: Introductionmentioning

confidence: 52%

Binary Representations of Algebras With at Most Two Binary Operations: A Cayley Theorem for Distributive Lattices

Movsisyan

2009

Int. J. Algebra Comput.

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“…The converse result, that every ternary algebra is isomorphic to a subset-pair algebra has been proved by Brzozowski, Lou, and Negulescu [4] for the finite case, and byÉsik [6] for the infinite case. Thus we have Theorem 3.…”

Section: Subset-pair Algebrasmentioning

confidence: 94%

“…Thus, by an involuted semilattice S, ∨, − , 1 we mean an algebra satisfying (1)- (6). b We refer to the unary operation − as quasi-complementation.…”

Section: Involuted Semilatticesmentioning

confidence: 99%

Involuted Semilattices and Uncertainty in Ternary Algebras

Brzozowski

2004

Int. J. Algebra Comput.

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An involuted semilattice S, ∨, − is a semilattice S, ∨ with an involution − : S → S, i.e., S, ∨, − satisfiesā = a, and a ∨ b =ā ∨b. In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice S, ∨, − , 1 with greatest element 1 is said to be complemented if it satisfies a ∨ā = 1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra T, +, * , − , 0, φ, 1 is a de Morgan algebra with a third constant φ satisfying φ =φ, and (a +ā) + φ = a +ā. If we define a third binary operation ∨ on T as a ∨ b = a * b + (a + b) * φ, then T, ∨, − , φ is a complemented semilattice.

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“…In this section, we take a more abstract view on the concept of "Cayley representation". In the literature (for example, [2,5,17,25]), authors usually define Cayley representations of different forms of algebraic structures in terms of embeddings. This means that given an object X, there is a homomorphism σ : X → Y to a different object Y , and moreover σ has a retraction (not necessarily a homomorphism) ρ : Y → X (meaning ρ · σ = id).…”

Section: Tight Cayley Representationsmentioning

confidence: 99%

Equational Theories and Monads from Polynomial Cayley Representations

Piróg

Polesiuk

Sieczkowski

2019

Lecture Notes in Computer Science

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We generalise Cayley's theorem for monoids by providing an explicit formula for a (multi-sorted) equational theory represented by the type P X → X, where P is an arbitrary polynomial endofunctor with natural coefficients. From the computational perspective, examples of effects given by such theories include backtracking nondeterminism (obtained with the original Cayley representation X → X), finite mutable state (obtained with n → X, for a constant n), and their different combinations (via n × X → X or X n → X). Moreover, we show that monads induced by such theories are implementable using the type formers available in programming languages based on a polymorphic λ-calculus, both as compositions of algebraic datatypes and as continuation-like monads. We give a set-theoretic model of the latter in terms of Barr-dinatural transformations. We also introduce CayMon, a tool that takes a polynomial as an input and generates the corresponding equational theory together with the two implementations of the induced monad in Haskell.

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A Cayley Theorem for Ternary Algebras

Abstract: We prove a representation theorem of ternary algebras by ternary functions.

Cited by 17 publications

References 4 publications

Binary Representations of Algebras With at Most Two Binary Operations: A Cayley Theorem for Distributive Lattices

Binary Representations of Algebras With at Most Two Binary Operations: A Cayley Theorem for Distributive Lattices

Involuted Semilattices and Uncertainty in Ternary Algebras

Equational Theories and Monads from Polynomial Cayley Representations

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