We give a characterization of nuclear Fréchet lattices in terms of lattice properties of the seminorms. Indeed, we prove that a Fréchet lattice is nuclear if and only if it is both an AL- and an AM-space.

In this paper, the variety of three-valued closure algebras, that is, closure algebras with the property that the open elements from a three-valued Heyting algebra, is investigated. Particularly, the structure of the finitely generated free objects in this variety is determined.

A subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function ƒ (D), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.

Let K and L be lattices, and let ϕ be a homomorphism of K into L.Then ϕ induces a natural 0-preserving join-homomorphism of Con K into Con L.

Extending a result of Huhn, the authors proved that if D and E are finite distributive lattices and ψ is a 0-preserving join-homomorphism from D into E, then D and E can be represented as the congruence lattices of the finite lattices K and L, respectively, such that ψ is the natural 0-preserving join-homomorphism induced by a suitable homomorphism ϕ: K → L. Let m and n denote the number of join-irreducible elements of D and E, respectively, and let k = max (m, n). The lattice L constructed was of size O(22(n+m)) and of breadth n+m.

We prove that K and L can be constructed as ‘small’ lattices of size O(k5) and of breadth three.

The covering relation in the lattice of subuniverses of a finite distributive lattices is characterized in terms of how new elements in a covering sublattice fit with the sublattice covered. In general, although the lattice of subuniverses of a finite distributive lattice will not be modular, nevertheless we are able to show that certain instances of Dedekind's Transposition Principle still hold. Weakly independent maps play a key role in our arguments.

If P is a partially ordered set and R is a commutative ring, then a certain differential graded R-algebra A•(P) is defined from the order relation on P. The algebra A•() corresponding to the empty poset is always contained in A•(P) so that A•(P) can be regarded as an A•()-algebra. The main result of this paper shows that if R is an integral domain and P and P′ are finite posets such that A•(P)≅A•(P′) as differential graded A•()-algebras, then P and P′ are isomorphic.

A unified study is undertaken of finitely generated varieties HSP () of distributive lattices with unary operations, extending work of Cornish. The generating algebra () is assusmed to be of the form (P; ∧, ∨, 0, 1, {fμ}), where each fμ is an endomorphism or dual endomorphism of (P; ∧, ∨, 0, 1), and the Priestly dual of this lattice is an ordered semigroup N whose elements act by left multiplication to give the maps dual to the operations fμ. Duality theory is fully developed within this framework, into which fit many varieties arising in algebraic logic. Conditions on N are given for the natural and Priestley dualities for HSP () to be essentially the same, so that, inter alia, coproducts in HSP () are enriched D-coproducts.

We investigate the number and size of the maximal sublattices of a finite lattice. For any positive integer k, there is a finite lattice L with more that ]L]k sublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

As a consequence of general principles, we add to the array of ‘hulls’ in the category Arch (of archimedean ℓ-groups with ℓ-homomorphisms) and in its non-full subcategory W (whose objects have distinguished weak order unit, whose morphisms preserve the unit). The following discussion refers to either Arch or W. Let α be an infinite cardinal number or ∞, let Homα; denote the class of α-complete homomorphisms, and let R be a full epireflective subcategory with reflections denoted rG: G → rG. Then for each G, there is rαG ∈ Homα (G, R) such that for each ϕ ∈ Homα (G, R), there is unique with . Moreover if every rG is an essential embedding, then, for every α and every G, rαG = rG, and every Homα. If and R consists of all epicomplete objects, then every Homw1. For α = ∞, and for any R, every Hom∞.

It is known that every frame is isomorphic to the generalized Gleason algebra of an essentially unique bi-Stonian space (X, σ, τ) in which σ is T0. Let (X, σ, τ) be as above. The specialization order ≤σ, of (X, σ) is τ × τ-closed. By Nachbin's Theorem there is exactly one quasi-uniformity U on X such that ∩U = ≤σ and J(U*) = τ. This quasi-uniformity is compatible with σ and is coarser than the Pervin quasi-uniformity U of (X, σ). Consequently, τ is coarser than the Skula topology of σ and coincides with the Skula topology if and only if U = P.

This paper studies higher dimensional analogues of the Tamari lattice on triangulations of a convex n-gon, by placing a partial order on the triangulations of a cyclic d-polytope. Our principal results are that in dimension d≤3, these posets are lattices whose intervals have the homotopy type of a sphere or ball, and in dimension d≤5, all triangulations of a cyclic d-polytope are connected by bistellar operations.

The random k-dimensional partial order Pk(n) on n points is defined by taking n points uniformly at random from [0,1]k. Previous work has concentrated on the case where k is constant: we consider the model where k increases with n.

We pay particular attention to the height Hk(n) of Pk(n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in Pk(n): if k – (t/log t!) log n tends to + ∞ then the probability that Pk(n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of Hk(n) for the entire range of k(n).

We also consider the maximum degree of Pk(n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k.

We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of Pk(n) for large k(n).

In this paper we introduce the notion of Riesz homomorphism on Archimedean directed partially ordered groups and use it to study the vector lattice cover of such groups.

A lattice-ordered power series algebra of a totally ordered field over a rooted abelian group may be constructed in a way that is arbitrary only in requiring that a factor set be chosen in the field and an extended total order be chosen on the group modulo its torsion subgroup. The resulting algebra is a field if and only if the subalgebra of elements with torsion support form a field. It follows that if the torsion subgroup may be independently embedded in the algebraic closure of the totally ordered field, or if the resulting algebra has no zero-divisors, then the algebra is a field. The set of supporting subsets for the power series may be characterized abstractly in such a way that previous representation theorems of lattice-ordered fields into power series algebras may be applied to produce representations into power series fields.

In this paper we consider classes of vector lattices over subfields of the real numbers. Among other properties we relate the archimedean condition of such a vector lattice to the uniqueness of scalar multiplication and the linearity of l-automorphisms. If a vector lattice in the classes considered admits an essential subgroup that is not a minimal prime, then it also admits a non-linear l-automorphism and more than one scalar multiplication. It is also shown that each l-group contains a largest archimedean convex l-subgroup which admits a unique scalar multiplication.

Any preorder P on a set X has an associated preorder P′, P″, P‴, … The proerties of this sequence are studied. When X is finite the sequence is eventually periodic with period P = 1 or p = 1, the eventual constant preorder is full p = 2 the possible forms which the eventual alternating order can take are examined: first, the possible combinations of components are enumerated; second, the notion of ramification at a caste is used to show that X may in a heuristic sense be of unbounded complexity. If X is orderdense the periodicity starts at P′.

In the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

In an earlier paper, we investigated for finite lattices a concept introduced by A. Slavik: Let A, B, and S be sublattices of the lattice L, A∩B = S, A∪B = L. Then L pastes A and B together over S, if every amalgamation of A and B over S contains L as a sublattice. In this paper we extend this investigation to infinite lattices. We give several characterizations of pasting; one of them directly generalizes to the infinite case the characterization theorem of A. Day and J. Ježk. Our main result is that the variety of all modular lattices and the variety of all distributive lattices are closed under pasting.

It is shown that every boolean right near-ring R is weakly commutative, that is, that xyz = xzy for each x, y, z ∈ R. In addition, an elementary proof is given of a theorem due to S. Ligh which states that a d.g. boolean near-ring is a boolean ring. Finally, a characterization theorem is given for a boolean near-ring to be isomorphic to a particular collection of functions which form a boolean near-ring with respect to the customary operations of addition and composition of mappings.

We characterize the generalized ordered topological spaces X for which the uniformity (X) is convex. Moreover, we show that a uniform ordered space for which every compatible convex uniformity is totally bounded, need not be pseudocompact.