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The Structure of Free Algebras 1 Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 85

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The Structure of Free Algebras 1 Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 85 The structure of free algebras 1 Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 South Morgan, Chicago, IL 60607, USA The structure of free algebras Joel BERMAN Abstract This article is a survey of selected results on the structure of free algebraic systems obtained during the past 50 years. The focus is on ways free algebras can be decomposed into simpler components and how the number of components and the way the components interact with each other can be readily determined. A common thread running through the exposition is a concrete method of representing a free algebra as an array of elements. 1 Introduction Let V be a variety (or equational class) of algebras. By FV (X) we denote the free algebra for V freely generated by the set X. The algebra FV(X) may be defined as an algebra in V generated by X that has the universal mapping property: For every A ∈V and every function v : X → A there is a homomorphism h : FV (X) → A that extends v, i.e., h(x) = v(x) for all x ∈ X. It is known that FV (X) exists for every variety V and is unique up to isomorphism. In this article we investigate the structure of free algebras in varieties. We present a very concrete, almost tactile, representation of FV(X) and some algebras closed related to FV(X) as a rectangular arrays of elements. The rows of such an array are indexed by the elements of the algebra and the columns are indexed by a set U of valuations, where a valuation is any function v from X to an algebra A ∈V for which the set v(X) generates all of A. This array is denoted Ge(X, U). In Section 1 we prove some general results about Ge(X, U). We are interested in finding small tractable sets U of valuations that can be used to represent FV (X). Several such U are described. The section also contains a detailed description of how a software package de- veloped by R. Freese and E. Kiss can be used to explicitly construct the array Ge(X, U) for finitely generated varieties V and finite X. The remaining three sections deal with decompo- sitions of FV(X) into well-behaved substructures, and how these substructures are organized among themselves in a systematic way. In Section 2 the substructures considered are con- gruence classes of the kernel of a canonical homomorphism of FV (X) onto FW (X) for W a well-behaved and well-understood subvariety of V. In Section 3 varieties V are described for which FV (X) can be decomposed into overlapping syntactically defined substructures that are very homogeneous and code, in various ways, families of finite ordered sets. Because of the way these subsets fit together, an inclusion-exclusion type of argument can be used to determine the cardinality of FV(X). The final section deals with direct decompositions of FV (X) into a product of directly indecomposable factors. Some specific general conditions on a variety V are provided that allow for the determination of the structure of the directly indecomposable factors and of the exact multiplicity of each factor in the product. 2 The structure of free algebras 3 For general facts about varieties, free algebras and universal algebra used in this paper the reader may consult [8] or [18]. For the most part we follow the notation and terminology found there. For a set X of variables and a variety V of similarity type τ, a typical term of type τ built from X is denoted t(x1,...,xn), where the xi are elements of X. If n and X are clear from the context we simply write t. The set of variables that actually appear in a term t A is denoted var(t). For a term t(x1,...,xn) and an algebra A in V, by t we denote the A n the term operation on A corresponding to t. Thus t : A → A and for a1,...,an ∈ A, A t (a1,...,an) is the element of A obtained by interpreting t to the fundamental operations of A and applying the resulting operation to a1,...,an. The universe of FV (X) is denoted FV(X) and we use the following notation for elements of FV(X): For x ∈ X we write x for FV(X) the element of FV(X) obtained by applying x to the generator x of FV (X) and for a FV (X) term t(x1,...,xn) we write t for t (x1,..., xn). Note that x = x for every x ∈ X, every element of FV(X) is of the form t for some term t, and that for distinct terms s and t we can have s = t. A crucial fact about free algebras used repeatedly throughout this paper is that V satisfies the identity s ≈ t for terms s and t if and only if the elements s and t are equal in FV(X). Written more succinctly this is the familiar V |= s ≈ t iff s = t. See, for example §11 of [8] for further details. For an algebra A and a set X we view each element v in the direct power AX as a function v : X → A. Given v ∈ AX we denote the algebra A by Alg(v). If v ∈ AX and A t(x1,...,xn) is a term in the variables of X, then v(t) denotes t (v(x1),...,v(xn)). We say v is a valuation if v(X) generates the algebra Alg(v). The set of all valuations from X to an algebra A is denoted val(X, A). For K a class of algebras, val(X, K) denotes the collection of all v ∈ val(X, A) for A ∈K. Valuations will play a central role in our exposition. If an algebra B is the direct product j∈J Bj , then we denote by prj the projection homomorphism from B onto Bj. For K ⊆ J the projection of B onto Bj is denoted Q j∈K prK. Q Let K be a set of algebras of the same similarity type, X a set, and U a subset of X (A : A ∈K). For x ∈ X let x ∈ v∈U Alg(v) denote the element given by prv(x) = v(x) for all v ∈ U. We let X = {x : x ∈ X}. The subalgebra of Alg(v) generated by X is S Q v∈U denoted Ge(X, U). Q 1.1 Lemma Let V be a variety and U ⊆ (AX : A ∈V). If val(X, V) ⊆ U, then Ge(X, U) is isomorphic to F (X). V S Proof The algebra Ge(X, U) is generated by X. The set U contains valuations to separate the elements of X, so X and X have the same cardinality. So it suffices to show that Ge(X, U) has the universal mapping property for X over V. Let w be any function from X into an algebra A ∈ V. Let v ∈ val(X, SgA(w(X))) be defined by v(x) = w(x) for every x ∈ X. By assumption v ∈ U. Then prv is a homomorphism from Ge(X, U) into A that extends w since prv(x) = v(x) = w(x). 2 1.2 Definition For a variety V and a set X, we call U ⊆ val(X, V) free for V if Ge(X, U) is isomorphic to FV (X). The set U is called independent for V if Ge(X, U) is isomorphic to v∈U Alg(v). Q 4 J. Berman We are very much interested in concrete representations of Ge(X, U) when U is free for V. To this end we first consider sufficient conditions on a set U ⊆ val(X, V) that force U to be independent. 1.3 Lemma Let V be a variety and suppose U ⊆ val(X, V) is such that for every pair of terms s and t for which V 6|= s ≈ t there exists v ∈ U such that v(s) 6= v(t). Then Ge(X, U) =∼ FV (X), that is, U is free for V. Proof By virtue of the previous lemma it suffices to show that the projection homomorphism prU from Ge(X, val(X, V)) onto Ge(X, U) is one-to-one. If s(x1,..., xn) and t(x1,..., xn) are distinct elements of Ge(X, val(X, V)), then V 6|= s ≈ t. So there exists v ∈ U for which v(s) 6= v(t). Then prv(s(x1,..., xn)) = s(v(x1),...,v(xn)) = v(s) 6= v(t) = t(v(x1),...,v(xn)) = prv(s(x1,..., xn)). Thus, prU (s(x1,..., xn)) 6= prU (t(x1,..., xn)). 2 1.4 Corollary Let V be a variety and X a set. (1) If VSI is the class of (finitely generated) subdirectly irreducible algebras in V and if U = val(X,VSI), then U is free for V. X (2) If V is generated by the finite algebras A1,..., Am and if U = (Ai : 1 ≤ i ≤ m), m |A |n then U is free for V and |F (n)| ≤ |Ai| i . V i=1 S The second part of this Corollary is presentedQ in Birkhoff’s 1935 paper [7]. 1.5 Definition Let V be an arbitrary variety and X a set. Given two valuations v and v0 with A = Alg(v) and A0 = Alg(v0) algebras in V, we say that v and v0 are equivalent, written v ∼ v0, if there exist homomorphisms h : A → A0 and h0 : A0 → A such that v0 = hv and v = h0v0.
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