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A partial lattice P is ideal-projective, with respect to a class C of lattices, if for every K ∈ C and every homomorphism φ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f : P → K for φ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to C. We prove the following: (1) A finite lattice P, belonging to a variety V, is sharply transferable with respect to V iff it is projective with respect to V and weakly distributive lattice homomorphisms, iff it is ideal-projective with respect to V, (2) Every finite distributive lattice is sharply transferable with respect to the class Rmod of all relatively complemented modular lattices, (3) The gluing D4 of two squares, the top of one being identified with the bottom of the other one, is sharply transferable with respect to a variety V iff V is contained in the varietyMω generated by all lattices of length 2, (4) D4 is projective, but not ideal-projective, with respect to Rmod , (5) D4 is transferable, but not sharply transferable, with respect to the variety M of all modular lattices. This solves a 1978 problem of G. Grätzer, (6)We construct a modular lattice whose canonical embedding into its ideal lattice is not pure. This solves a 1974 problem of E. Nelson. [ABSTRACT FROM AUTHOR]

Abstract: Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam [D. Maharam, An algebraic characterization of measure algebras, Ann. of Math. (2) 48 (1947) 154–167] in relation to Von Neumann’s problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the main open problem in this area for around 60 years. It was finally resolved by Talagrand [M. Talagrand, Maharam’s problem, preprint, 31 pages, 2006] who provided the first example of a Maharam algebra which is not a measure algebra. In this paper we survey some recent work on Maharam algebras in relation to the two conditions proposed by Von Neumann: weak distributivity and the countable chain condition. It turns out that by strengthening either one of these conditions one obtains a ZFC characterization of Maharam algebras. We also present some results on Maharam algebras as forcing notions showing that they share some of the well-known properties of measure algebras. [Copyright &y& Elsevier]

Abstract: We construct an algebraic distributive lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R.P. Dilworth, from the forties. The lattice D has a compact top element and compact elements. Our results extend to any algebra possessing a congruence-compatible structure of a join-semilattice with a largest element. [Copyright &y& Elsevier]

Abstract: The paper is devoted to an algebraic and geometric study of the feasible set of a poset, the set of finite probability distributions on the elements of the poset whose weights satisfy the order relationships specified by the poset. For a general poset, this feasible set is a barycentric algebra. The feasible sets of the order structures on a given finite set are precisely the convex unions of the primary simplices, the facets of the first barycentric subdivision of the simplex spanned by the elements of the set. As another fragment of a potential complete duality theory for barycentric algebras, a duality is established between order-preserving mappings and embeddings of feasible sets. In particular, the primary simplices constituting the feasible set of a given finite poset are the feasible sets of the linear extensions of the poset. A finite poset is connected if and only if its barycentre is an extreme point of its feasible set. The feasible set of a (general) disconnected poset is the join of the feasible sets of its components. The extreme points of the feasible set of a finite poset are specified in terms of the disjointly irreducible elements of the semilattice presented by the poset. Semilattices presented by posets are characterised in terms of various distributivity concepts. [Copyright &y& Elsevier]