When the standard representation of a crystallographic Coxeter group Γ is reduced modulo an odd prime p, one obtains a finite group G acting on some orthogonal space over Zp. If Γ has a string diagram, then G p will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to gr...

Suppose the group G is generated by permutations gvg2,...,g, acting on a set CI of size n, such that 8182--8, ' s the identity permutation. If the generator gt has exactly ci cycles (for 1 ^ / < s), and G is transitive on Ci, then n(s—2)—YJ'(-I c, + 2 is a non-negative even integer. This is proved using an elementary graph-theoretic argument. In this paper we give a simple proof of a necessary ...

This paper presents a graph-based knowledge representation and reasoning language. This language benefits from an important syntactic operation, which is called a graph homomorphism. This operation is sound and complete with respect to logical deduction. Hence, it is possible to do logical reasoning without using the language of logic but only graphical, thus visual, notions. This paper present...

Structural join has been established as a primitive technique for matching the binary containment pattern, specifically the parent–child and ancestor–descendant relationship, on the tree XML data. While current indexing approaches and evaluation algorithms proposed for the structural join operation assume the tree-structured data model, the presence of reference links in XML documents may rende...

A new type of decomposition of graphs – the operator decomposition connected with the classical notion of homogeneous set (or module) is introduced. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operati...

This paper presents a graph-based knowledge representation and reasoning language. This language benefits from an important syntactic operation, which is called a graph homomorphism. This operation is sound and complete with respect to logical deduction. Hence, it is possible to do logical reasoning without using the language of logic but only graphical, thus visual, notions. This paper present...

We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs 〈L1,L2〉 of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to L1 and L2, respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary a...

When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d ≥ 2, one obtains a finite group G which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, we here develop methods to handle composite moduli and completely describe the corresponding modular polyt...

In this article we study the modular product graph operation, denoted by ♦, with particular emphasis on when the operation preserves the property of a graph being 3-existentially closed. We characterise when G♦H is 3e.c. provided that H is 3-e.c., and we establish two new infinite classes of 3-e.c. graphs of the form G♦H, for which G need not be 3-e.c. itself.