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We define in this paper a certain notion of completeness for a wide class of commutative (pre)ordered monoids (from now on P.O.M.'s). This class seems to be the natural context for studying structures like measurable function spaces, equidecomposability types of spaces, partially ordered abelian groups and cardinal algebras. Then, we can prove that roughly speaking, spaces of measures with values in complete P.O.M.'s are complete P.O.M.'s. Furthermore, this notion of completeness yields us an 'arithmetical' characterization of injective P.O.M.'s.

No abstract

For a partially ordered set P , we denote by Co(P ) the lattice of order-convex subsets of P . We find three new lattice identities, (S), (U), and (B), such that the following result holds.Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co(P ) iff L satisfies (S), (U), and (B).Furthermore, if L has an embedding into some Co(P ), then it has such an embedding that preserves the existing bounds. If L is finite, then one can take P finite, withwhere J(L) denotes the set of all join-irreducible elements of L.On the other hand, the partially ordered set P can be chosen in such a way that there are no infinite bounded chains in P and the undirected graph of the predecessor relation of P is a tree.

Let A and B be lattices with zero. The classical tensor product, A ⊗ B, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: A ⊗ B is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice.Let Conc L denote the join-semilattice with zero of compact congruences of a lattice L. Our main result is that the following isomorphism holds for any capped tensor product: