Algebra Universalis最新文献

We consider S-operations (f :A^{n} rightarrow A) in which each argument is assigned a signum (s in S) representing a “property” such as being order-preserving or order-reversing with respect to a fixed partial order on A. The set S of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of S-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all S-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of S-preclone. We introduce S-relations (varrho = (varrho _{s})_{s in S}), S-relational clones, and a preservation property (), and we consider the induced Galois connection ({}^{S}{}textrm{Pol})–({}^{S}{}textrm{Inv}). The S-preclones and S-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all S-preclones on A.

We define a special network that exhibits the large embeddings in any class of similar algebras. With the aid of this network, we introduce a notion of distance that conceivably counts the minimum number of dissimilarities, in a sense, between two given algebras in the class in hand; with the possibility that this distance may take the value (infty ). We display a number of inspirational examples from different areas of algebra, e.g., group theory and monounary algebras, to show that this research direction can be quite remarkable.

The N-star compactifications of frames are the frame-theoretic counterpart of the N-point compactifications of locally compact Hausdorff spaces. A (pi )-compactification of a frame L is a compactification constructed using a special type of a basis called a (pi )-compact basis; the Freudenthal compactification is the largest (pi )-compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all N-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which N-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification.

We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We we give two general constructions of unilinear residuated lattices, provide an axiomatization and a proof-theoretic calculus for the variety they generate, and prove the finite model property for various subvarieties.

Let X be an Archimedean vector lattice and (X_{+}) denote the positive cone of X. A unary operation (varpi ) on (X_{+}) is called a truncation on X if

Let (X^{u}) denote the universal completion of X with a distinguished weak element (e>0.) It is shown that a unary operation (varpi ) on (X_{+}) is a truncation on X if and only if there exists an element (uin X^{u}) and a component p of e such that

Here, px is the product of p and x with respect to the unique lattice-ordered multiplication in (X^{u}) having e as identity. As an example of illustration, if (varpi ) is a truncation on some (L_{p}left( {mu } right) )-space then there exists a measurable set A and a function (uin L_{0}left( {mu } right) ) vanishing on A such that (varpi left( xright) =1_{A}x+uwedge x) for all (xin L_{p}left( {mu } right) .)

In this paper, new purely topological approaches are furnished in order to characterize self-majorizing elements in an Archimedean vector lattice A. More precisely, it is shown that an element (0<fin A) is a self-majorizing element if and only if every f-maximal order ideal of A is relatively uniformly closed. In addition, it is proved that self-majorizing elements are characterized via the hull–kernel topology on both the set of all proper prime order ideals ({mathcal {P}}) and on the set of all g-maximal order ideals ({mathcal {Q}}) of A,  for all (gin A^{+}.) In fact, the set of all prime order ideals of A not containing f (respectively, the set of all g-maximal order ideals of A not containing f,  for all (gin A^{+})) is a closed with respect to the hull–kernel topology on ({mathcal {P}}) (respectively, on ({mathcal {Q}})) if and only if f is a self-majorizing element in A.

We prove that slim patch lattices are exactly the absolute retracts with more than two elements for the category of slim semimodular lattices with length-preserving lattice embeddings as morphisms. Also, slim patch lattices are the same as the maximal objects L in this category such that (|L|>2.) Furthermore, slim patch lattices are characterized as the algebraically closed lattices L in this category such that (|L|>2.) Finally, we prove that if we consider ({0,1})-preserving lattice homomorphisms rather than length-preserving ones, then the absolute retracts for the class of slim semimodular lattices are the at most 4-element boolean lattices.

We give a self-contained proof of the following result: Finitely additive probability measures (also known as “states”) of the free boolean algebra ({mathsf F}_omega ) over the free generating set ({X_1,X_2,ldots }) having the invariance property under finite permutations of the (X_i), coincide with states lying in the closure of the set of convex combinations of product states of ({mathsf F}_omega ) in the vector space (mathbb R^{{mathsf F}_omega }) equipped with the product topology. De Finetti’s celebrated exchangeability theorem can be easily recovered from our proof.

We solve some problems about relative lengths of Maltsev conditions, in particular, we provide an affirmative answer to a classical problem raised by A. Day more than 50 years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms (t_0,ldots , t_n) witnessing congruence distributivity it is possible to construct terms (u_0,ldots , u _{2n-1} ) witnessing congruence modularity. We show that Day’s result about the number of such terms is sharp when n is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, as well as with possible variations we will call “specular” and “defective”. All the results hold when restricted to locally finite varieties.

We study two measures of associativity for graph algebras of finite undirected graphs: the index of nonassociativity and (a variant of) the semigroup distance. We determine “almost associative” and “antiassociative” graphs with respect to both measures. It turns out that the antiassociative graphs are exactly the balanced complete bipartite graphs, no matter which of the two measures we consider. In the class of connected graphs the two notions of almost associativity are also equivalent.