Universal Algebra

Vyacheslav A. Artamonov, Günter F. Pilz, Boris I. Plotkin, Kalle Kaarli, Lev N. Shevrin, Evgeny V. Sukhanov, Mikhail V. Volkov, A. G. Pinus, Yefim Katsov, Leonid Bokut’, Hajnal Andréka, Judit X. Madarász, István Németi, Peter Burmeister and Hans-Dieter Ehrich

Chapter Chapter G in The Concise Handbook of Algebra, 2002, pp 451-490 from Springer

Abstract: Abstract An n -ary operation (n ∈ ℕ0) on a set A is a map ω: A n → A, where A 0 := { ∅ }. The number n is called the arity of ω. A universal algebra is a pair 𝒜 = (A, Ω) consisting of a non-empty set A and a set Ω of operations on A. The set A and the members of Ω are called the universe and the fundamental operations of the algebra 𝒜, respectively. In practice, one usually is not interested in a single, isolated algebra but in a class of algebras of the same type. Therefore it is more customary to consider the set Ω not as the set of operations on the given set A but rather as the set of operation symbols. Formally this is achieved by first introducing the notion of type. The type is a set Ω together with a partition Ω = Ω0 ∪ Ω1 ∪ Ω2⋯. (Empty Ω i are allowed.) Alternatively, the type is a set Ω together with a mapping r: Ω → ℕ0. A universal algebra of type Ω, or simply an Ω-algebra is a pair 𝒜 = (A, Ω) where A is a non-empty set and to every ω ∈ Ω n it is assigned an n-ary operation on A, denoted by the same symbol ω. Cf. Section G.10 for a generalization of these concepts.

Keywords: Word Problem; Universal Algebra; Free Algebra; Term Operation; Partial Algebra (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-94-017-3267-3_7

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