According to the Sees Theorem, every completely 0-simple semigroup can be represented by a Rees matrix semigroup over a group with zero. A characterization of all subsemigroups of the latter is given in terms of the structure group, structure sets, and two mappings. Next all congruences on such subsemigroups are described, along with conditions for comparability. Finally, an algorithm for computing the number of nonisomorphic inverse subsemigroups is constructed.

In this paper we introduce the concept of a δθ-cover to generalize Aquaro's Theorem that every point countable open cover of a topological space such that every discrete closed family of sets is countable has a countable subcover. A δθ-cover of a space X is defined to be a family of open sets where each Vn covers X and for x є X there exists n such that Vn is of countable order at x. We replace point countable open cover by a δθ-cover in Aquaro's Theorem and also generalize the result of Worrell and Wicke that a θ-refinable countably compact space is compact and Jones′ result that ℵ1-compact Moore space is Lindelöf which was used to prove his classic result that a normal separable Moore space is metrizable, using the continuum hypothesis.

Let S be a topological semigroup with separately continuous multiplication and H a uniformly closed invariant subspace of LUC(S) (the space of left uniformly continuous bounded functions on S ) that contains the constants. It is shown that if H is left introverted and H admits a tight two-sided invariant mean m, then for each h ∈ H, m(h) is the unique constant function in the norm closed convex hull of the left orbit of h; consequently, H has a unique left invariant mean. (In fact, it is enough for H to admit a tight right invariant mean and a left invariant mean. ) For certain S, a similar result is obtained when H is a left compact-open introverted subspace of LCC(S) (the space of left compact-open continuous functions on S ).

We give the representation formula of a periodic function with period one in terms of its translated means on the unit interval. We also give an application of this formula to a boundary value problem.

Let d(G) denote the minimum of the cardinalities of the generating sets of the group G. Call a generating set of cardinality d(G) a minimal generating set for G. If A is a finitely generated nilpotent group, B a non-trivial finitely generated abelian group and A wr B is their (restricted, standard) wreath product, then it is proved (by explicitly constructing a minimal generating set for A wr B ) that d(AwrB) = max{l+d(A), d(A×B)} where A × B is their direct product.

The remark is that the recent work in this Bulletin dealing with the extension of a partial to a total strict order on a module over a partially ordered ring falls under standard order extension results for operator groups once it is noted that the added requirement of strictness just comes to injective operation of the positive scalars. These standard results are in turn generalized to a universal algebra setting.

Results of Tutte and of Anderson giving conditions for a simple graph G to have a perfect matching are generalized to give conditions for G to have a matching of defect d. A corollary to one of these results is a theorem of Berge on the size of a maximum matching in G.