If F is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.

We give a short, direct proof that given any finite group G there exist positive integers k and l and partitions α1 and α2 of {1, . . . , kl} into l subsets of size k such that (Skl)α1,α2 ∼= G. The method used will also show that given any finite group G there exists a regular bipartite graph whose automorphism group is isomorphic to G.

Let G := (V,E) be a simple graph; for I ⊆ V we denote by l(I) the number of components of G[I], the subgraph of G induced by I. For V1, . . . , Vn finite subsets of V , we define a function β(V1, . . . , Vn) which is expressed in terms of l `Sn i=1 Vi ́ and l(Vi ∪ Vj) for i ≤ j. If V1, . . . , Vn are pairwise disjoint finite independent subsets of V , the number β(V1, . . . , Vn) can be computed...

The meq gene was thought to be only detected in Marek's disease virus serotype 1 (MDV 1) including a very virulent strain, Md5, while L-meq, in which a 180-bp sequence is inserted into the meq open reading frame, is found in other strains of MDV 1, such as CVI 988/R6. However, both meq and L-meq were previously detected by PCR in chickens infected with MDV 1, suggesting that MDV 1 may consists ...

Given the integers 1,, k,,1 2 , k 2 , r, which satisfy the condition 1,, 1 2 > r> k,, k 2 > 0, we define m = N(1,, k, ; 1 2 , k 2 ; r) as the smallest integer with the following property : if S is a set containing m points and the r-subsets of S are partitioned arbitrarily into two classes, then for i = 1 or 2 there exists an l i subset of S each of whose ki-subsets lies in some r-subset of the...

We give a very simple new proof of the celebrated intersection theorem of D. K. Ray-Chaudhuri and R. M. Wilson. The new proof yields a generalization to nonuniform set systems. Let N(n, s, r) = ( n s ) + ( n s− 1 ) + · · ·+ ( n s− r + 1 ) . Generalized Ray-Chaudhuri – Wilson Theorem. Let K = {k1, . . . , kr}, L = {l1, . . . , ls}, and assume ki > s − r for all i. Let F be a family of subsets of...

We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordina...

Phagocytes such as dendritic cells and macrophages, which are distributed in the small intestinal mucosa, play a crucial role in maintaining mucosal homeostasis by sampling the luminal gut microbiota. However, there is limited information regarding microbial uptake in a steady state. We investigated the composition of murine gut microbiota that is engulfed by phagocytes of specific subsets in t...

We call a subset of an ordinal recognizable if it is the unique subset x of for which some Turing machine with ordinal time and tape and an ordinal parameter, that halts for all subsets of as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given ⌃1 formula in L[x]. We further define the recognizable closure for subsets of by closing under r...

A kind of new fuzzy compactness, called near SR-compactness, is de4ned for an arbitrary L-subset and for L a complete DeMorgan algebra. This kind of fuzzy compactness is between Lowen’s strong fuzzy compactness and the second author’s SR-compactness. Characterizations and examples of near SR-compact L-subsets are given, and its topological properties are systematically investigated. c © 2003 El...