Algebra Universalis最新文献

We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence. The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even one. We give a combinatorial proof of the lattice property, relying on a description of words for the Garside element in terms of Schröder trees, and on a recursive description of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins in the lattice. As a byproduct we also obtain that the number of words for the Garside element is given by a little Schröder number.

In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories—whose models are sets equipped with an action by a monoid M—and all hyperaffine theories—whose models are sets with an action by a Boolean algebra B. We improve on Johnstone’s result by showing that an equational theory is cartesian closed just when its operations have a unique hyperaffine–unary decomposition. It follows that any non-degenerate cartesian closed variety is a variety of sets equipped with compatible actions by a monoid M and a Boolean algebra B; this is the classification theorem of the title.

We provide a simple natural duality for the varieties generated by the negation- and implication-free reduct of a finite MV-chain. We study these varieties through the dual equivalences thus obtained. For example, we fully characterize their algebraically closed, existentially closed and injective members. We also explore the relationship between this natural duality and Priestley duality in terms of distributive skeletons and Priestley powers.

A quasivariety (mathfrak N) is called relative congruence principal if, for every algebra (Ain mathfrak N), every compact (mathfrak N)-congruence on A is a principal (mathfrak N)-congruence. We characterize relative congruence principal quasivarieties in terms of one identity and two quasi-identities. We will use the characterization to show that there exists a continuum of relative congruence principal quasivarieties of algebras of a signature (sigma ), provided (sigma ) contains at least one operation of arity greater than 1. Several examples are provided.

The override operation (sqcup ) is a natural one in computer science, and has connections with other areas of mathematics such as hyperplane arrangements. For arbitrary functions f and g, (fsqcup g) is the function with domain ({{,textrm{dom},}}(f)cup {{,textrm{dom},}}(g)) that agrees with f on ({{,textrm{dom},}}(f)) and with g on ({{,textrm{dom},}}(g) backslash {{,textrm{dom},}}(f)). Jackson and the author have shown that there is no finite axiomatisation of algebras of functions of signature ((sqcup )). But adding operations (such as update) to this minimal signature can lead to finite axiomatisations. For the functional signature ((sqcup ,backslash )) where (backslash ) is set-theoretic difference, Cirulis has given a finite equational axiomatisation as subtraction o-semilattices. Define (fcurlyvee g=(fsqcup g)cap (gsqcup f)) for all functions f and g; this is the largest domain restriction of the binary relation (fcup g) that gives a partial function. Now (fcap g=fbackslash (fbackslash g)) and (fsqcup g=fcurlyvee (fcurlyvee g)) for all functions f, g, so the signatures ((curlyvee )) and ((sqcup ,cap )) are both intermediate between ((sqcup )) and ((sqcup ,backslash )) in expressive power. We show that each is finitely axiomatised, with the former giving a proper quasivariety and the latter the variety of associative distributive o-semilattices in the sense of Cirulis.

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) .)