Showing papers in "Algebra Universalis in 1977"

Boolean algebras as unions of chains of subalgebras

Abstract: where, for a < h, B,~ is a subalgebra of B and a No. Let B = {b~ I a < )t} and define, for a < h, B~ to be the subalgebra of B generated by {b~[~< a}. Each B~ is a proper subalgebra of B (for h = No, this holds since a finitely generated BA is finite), and thus, cf(B)-< h = card(B). There are some very simple facts:

On the existence of subalgebras of direct products with prescribedd-fold projections

Abstract: Baker and Pixley have shown the equivalence of a number of conditions on a positive integerd and a varietyV be uniquely determined by its projections in thed-fold subproductsA i (1)×...×A i (d). It is shown here thatunder Baker and Pixley's conditions this uniqueness result is complemented by an existence result: SupposeA 1,...,A r ∈V, and that for everyd-tupleI={i(1),...,i(d)} a subalgebraS I ⊆A i(1)×...×A i(d) is given. Then these data are the projections of one subalgebraS⊆ A 1×...A r if and only if they are “consistent” on eachd+1-tuple {i(1),...,i(d+1)}. In the case where eachA i is the lattice {0, 1}, these results lead to the well-known description of finite distributive lattices in terms of finite partially ordered sets. Under appropriate hypotheses the above result generalizes to subalgebras of infinite direct products.

An application of Whitman's condition to lattices with no infinite chains

Abstract: Let K be a class of lattices. A lattice L ~ K is projective in K if, for each epimorphism 4~:A--~B between lattices A and B in K, each homomorphism ~b:L--->B lifts to a homomorphism $ ' : L * A such that $'~b = ~. A lattice L is a retract of a lattice M if there are homomorphisms q~.: M--> L and tp:L--~ M such that $~b = idL; by necessity ~b is an epimorphism and $ is an embedding. We shall say that condition Qx(L) holds if for each M ~ K and every epimorphism ~b :M--~ L there exists a homomorphism $ : L---~ M such that $~b = idL. A simple universal-algebraic argument establishes the following lemma.

Embedding and unsolvability theorems for modular lattices

Abstract: LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ℳ be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ℒ(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω. THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ⊂ ℳ, such thatL(R, ω) is inV. THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ*) having five generators, where δ*=δ for infinite δ and δ*=4δ(m+1) if δ is finite andL has a set ofm generators. THEOREM. The free ℒ(R)-lattice on ω generators is embeddable in the free ℒ(R)-lattice on five generators. THEOREM. IfL has an (m, n), ℒ(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ℒ(R)-presentation.

Equational theory of algebras with a majority polynomial

Abstract: This note gives a simple proof of a result of R. N. McKenzie that any finitely based variety of algebras which admit a majority polynomial and is definable by absorption identities is one-based. In the case of lattices, for example, this yields a “short” identity with only seven variables.

An example concerning definable principal congruences

Abstract: In [1] Baldwin and Berman ask if a variety generated by a finite algebra 91 has definable principal congruences, i.e. is there a first-order formula ~(x, y, u, v) such that for any ~eo//.(91) and a , b , c , d in 2~ we have (c, d>~O(a,b) iff ~cI)(a, b, c, d). In the following we describe a four-element algebra 91 such that ~ has distributive congruence lattices and does not have definable principal congruences. Let 91 =

Injective and projective zero-dimensional compact universal algebras

Abstract: This paper firstly deals with injectivity in the class of all zero-dimensional compact associative and distributive universal algebras. In [9, 5], it is proved that every injective zero-dimensional compact semi-lattice (or distributive lattice or Boolean algebra) is a retract of a product of finite injectives. In other words, every injective in the subclass Fs~ of finite algebras is also an injective in the class KeN of all zero-dimensional compact ones. In [1], many properties in the class Fs~, which are preserved in the class of all profinite algebras, are obtained. What is to be discussed here is that if the given algebras are all associative and distributive (see definitions in [4]), then the injectivity in Fs~ is a property that is preserved in the class KeN. Secondly, we obtain a necessary and sufficient condition that all projective compact algebras in the class Ksd of all compact algebras to be zero-dimensional, provided that the underlying algebra of any projective one in Ks~ is residually finite. The terminologies and definitions used here mainly follow those in [8] for universal algebras and those in [1, 4] for the others. Throughout this paper, all universal algebras will be of finite type, i.e., by a ,r-algebra we mean a pair (A, (f~)i~), where A is a set, (fi)i~ a family of operations: maps f~ of A "~ into A, where m~ is the arity of the fi, that is a non-negative integer and I is a finite ordered set {1, 2 , . . . , t}. Then we say that it is of arity type z = (rnx, m2, 9 9 9 mr). And all ~--algebras in this paper are assumed to be associative and distributive (see [4]). A topological ,r-algebra is an object (A, (f~)~i, 5-), where (A, (f~)i~i) is a T-algebra and 5 is a Hausdorff topology on A such that each fi is a continuous map of the product space (Am', 5"m'.) into (A,5").

The algebraic closure of a semigroup of functions

Abstract: A semigroup of functionsS υAA isalgebraic providedA can be endowed with some collection of (finitary) operations to produce an algebraU withS = EndU, the endomorphisms ofU. Thealgebraic closure ofS is\(\bar S\)=End,Us, whereUs is the algebra of all finitary operations which admit eachf σS as a homomorphism. Here we prove, for A finite, thatg σ\(\bar S\) iffg is the unique solution to a system ζ of functional equations each of the formfx=h orfx=y with, coefficientsf, h σS. For A infinite a similar local condition holds. Applications to related problems are given.

The algebraic closure of a function

Abstract: The algebraic operations on a set A which admit (as a homomorphism) a particular mapf∈AA, admit all powers off and certain other maps as well The algebraic closure off,\(\bar f\), is the totality of all maps which admit the algebraic operations thatf does The purpose of this paper is to characterize\(\bar f\) in terms off, both forA finite and forA infinite