By using the concept of tame embeddings of chains, a characterization is given of the subobjects of the lattice-ordered groups of order-automorphisms of the chains of rational and real numbers.

The concept of ‘Almost Distributive Lattices’ (ADL) is introduced. This class of ADLs includes almost all the existing ring theoretic generalisations of a Boolean ring (algebra) like regular rings, P-rings, biregular rings, associate rings, P1-rings, triple systems, etc. This class also includes the class of Baer-Stone semigroups. A one-to-one correspondence is exhibited between the class of relatively complemented ADLs and the class of Almost Boolean Rings analogous to the well-known Stone's correspondence. Many concepts in distributive lattices can be extended to the class of ADLs through its principal ideals which from a distributive lattice with 0. Subdirect and Sheaf representations of an ADL are obtained.

The lateral completion of a completely distributive lattice-ordered permutation group is investigated via various completions, obtained by adjoining permutations which match some elements of the given group in various ways. This makes known results on the lateral completion of a completely distributive lattice-ordered group both transparent and easy.

Reduced rings and lattice-ordered groups are examples of groups with Boolean orthogonalities. In this note we show that any group with a Boolean orthogonality satisfying a finiteness condition introduced by Stewart is isomorphic with a group of homeomorphisms of a topological space, in which two homeomorphisms are orthogonal if and only if they have disjoint supports.

Let Q be the ring of quotients of the f-ring R with respect to a positive hereditary torsion theory and suppose Q is a right f-ring. It is shown that if the finitely-generated right ideals of R are principal, then Q is an f-ring. Also, if QR is injective, Q is an f-ring if and only if its Jacobson radical is convex. Moreover, a class of po-rings is introduced (which includes the classes of commutative po-rings and right convex f-rings) over which Q(M) is an f-module for each f-module M.

A semilattice tree T with 0 is slim if there is a chain C with 0 so that the lattices θ (T) and θ(C) of semilattice congruences are isomorphic. This paper establishes elementary consequences of slimness and uses simple constructive techniques to show certain small trees slim. If T is the union of at most countably many branches, each of which has a maximum or a countable cotinal subset, then T is slim. For trees with enough maximals slimness is equivalent with not having any uncountable anti-chains. If a tree T has a countable cofinal subset then T is slim. Thus finitary trees are slim.

Epstein and Horn, in their paper ‘Chain based lattices’, characterized P1-lattices, and P2-lattices in terms of their prime ideals. But no such prime ideal characterization for P0-lattices was given. Our main aim in this paper is to characterize P0-lattices in terms of their prime ideals. We also give a necessary and sufficient condition for a P-algebra to be a P0-lattice (and hence a P2-lattice).

If L is any semilattice, let TL denote the Munn semigroup of L, and Aut (L) the automorphism group of L.

We show that every semilattice L can be isomorphically embedded as a convex subsemilattice in a semilattice L' which has a transitive automorphism group in such a way that (i) every partial isomorphism α of L can be extended to an automorphism of L', (ii) every partial isomorphism: α: eL → fL of L can be extended to a partial isomorphism αL′: eL′ → fL′ of L′ such that TL → TL′, α → αL′ embeds TL' isomorphically in TL′, (iii) every automorphism γ of L can be extended to an automorphism γL′ of L′ such that Aut (L) → Aut (L′), γ → γL embeds Aut (L) isomorphically in Aut (L′).

Let δ be a root system and let V be the Hahn group of real-valued functions on δ Then δ can be order-embedded into P(δ), the root system of prime l-ideals of V. In this note we identify P(δ) in terms of δ without explicit reference to V, up to the convex subgroup structure of the additive groups of real closed η1-fields. In particular, we characterize the minimal prime 1-ideals of V in terms of δ by an ultrafilter construction which generalizes the well-known method when δ is trivially ordered.

Let P be a partially-ordered set in which every two elements have a common lower bound. It is proved that there exists a lower semilattice L whose elements are labelled with elements of P in such a way that (i) comparable elements of L are labelled with elements of P in the same strict order relation; (ii) each element of P is used as a label and every two comparable elements of P are labels of comparable elements of L; (iii) for any two elements of L with the same label, there is a label-preserving isomorphism between the corresponding principal ideals. Such a structure is called a full, uniform P-labelled semilattice.

An attempt is made to extend the theory of extensions of partial orders in groups to strict partially ordered N-groups. Necessary and sufficient conditions, for a strict partial order of an N-group to have a strict full extension, and for a strict partial order of an N-group to be an intersection of strict full orders, are obtained when the partially ordered near-ring N and the N-group G satisfy the condition (− x) n = − xn for all elements x in G and positive elements n in N.

A totally ordered set (and corresponding order-type) is said to be rigid if it is not similar to any proper initial segment of itself. The class of rigid ordertypes is closed under addition and multiplication, satisfies both cancellation laws from the left, and admits a partial ordering that is an extension of the ordering of the ordinals. Under this ordering, limits of increasing sequences of rigid order-types are well defined, rigid and satisfy the usual limit laws concerning addition and multiplication. A decomposition theorem is obtained, and is used to prove a characterization theorem on rigid order-types that are additively prime. Wherever possible, use of the Axiom of Choice is eschewed, and theorems whose proofs depend upon Choice are marked.