Conference on Ordered Algebraic Structures

For n a positive integer, Ln is the variety of `-groups satisfying the equation [x, y] = e. Many of the well known but striking features of the Ln varieties carry over to varieties satisfying the laws [x, y, z] = e and [x, y, y] = e, such as the structure of finitely generated subdirectly irreducible `-groups and the respective intersections with the representable variety. Some interesting examples of subdirectly irreducible `-groups that are neither nilpotent nor Scrimger will be shown, with discussion of their varieties. Commutative rings in which components of zero are essential Themba Dube Abstract For a prime ideal P of a commutative ring A with identity, we denote (as usual) by OP its component of zero, that is, the set of those members of P which are annihilated by non-members of P . We say A is essentially good if OP is an essential ideal whenever P is essential. I will present a characterization of these rings in terms of properties of their frames of radical ideals. I will give an outline of the proof that the direct product of any collection of essentially good rings is essentially good. The ring C(X) is essentially good if and only if the underlying set is infinite. Replacing OP with the pure part of P , we obtain a stronger variant of essential goodness, which is still characterizable in terms of the frame of radical ideals. The σ-property in C(X) Anthony W. Hager Abstract A vector lattice is a real linear space with a compatible lattice-order. Examples are C(X) (continuos functions from the topological space X to the reals), and any abstract ”measurable functions mod null functions”. The σ-property of a vector lattice A is (s) For each sequence {a(n)} in A,there are a sequence {p(n)} of positive reals and a ∈ A for which p(n)a(n) < a for each n. C(X), for X compact (trivial); Lebesgue Measurable functions mod Null (not trivial–connected with Egoroff’s Theorem). An application: If a quotient A/I has (s), then the quotient map lifts disjoint sets to disjoint sets. Here,we consider which C(X) have (s). For example: For discrete X, C(X) has (s) iff the cardinality of X < the bounding number b. For metrizable X, C(X) has (s) iff X is locally compact and each open cover has a subcover of size < b. (This much studied ”b” is the minimum among cardinals m for which each family of functions from the positive integers N to N of size m is bounded in the order of eventual domination for such functions. It is uncountable, no bigger than c, and regular. In ZFC, not much more can be said.) Higher order z-ideals in commutative rings Oghenetega Ighedo*, T. Dube