Reinhard Pöschel

Galois Connections for Operations and Relations

This paper reports on various Galois connections between operations and relations. Several specifications and generalizations are discussed.

MINIMAL PARTIAL CLONES

Let A be a finite set. A partial clone on A is a composition closed set of operations containing all projections. It is well known that the partial clones on A, ordered by inclusion, form a lattice. We show that the minimal partial clones on A are: (a) minimal clones of full operations or (b) generated by partial projections defined on a totally reflexive and totally symmetric domain.

Graph algebras and graph varieties

Graph algebras establish a connection between graphs (i.e. binary relations) and universal algebras. A structure theorem of Birkhoff-type is given which characterizes graph varieties, i.e. classes of graphs which can be defined by identities for their corresponding graph algebras: A class of finite directed graphs without multiple edges is a graph variety iff it is closed with respect to finite restricted pointed subproducts and isomorphic copies. Several applications are given, e.g., every loopless finite directed graph is an induced subgraph of a direct power of a graph with three vertices. Graphs with bounded chromatic number or density form graph varieties characterizable by identities of special kind.

Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems

Abstract The set of circulant graphs with p k vertices ( k⩾1, p an odd prime) is decomposed into a collection of well-specified subsets. The number of these subsets is equal to the k th Catalan number, and they are in one-to-one correspondence with the monotone underdiagonal walks on the plane integer (k+1)×(k+1) lattice. The counting of non-isomorphic circulant graphs in each of the subsets is presented as an orbit enumeration problem of Polya type with respect to a certain Abelian group of multipliers. The descriptions are given in terms of equalities and congruences between multipliers in accordance with an isomorphism theorem for such circulant graphs. In this way, explicit uniform counting formulae have been obtained for various types of circulant graphs with p 2 vertices.

Non-Cayley-isomorphic self-complementary circulant graphs

Several results concerning existence of k-paths, for which the sum of their vertex degrees is small, are presented.

A Short Introduction to Clones

In universal algebra, clones are used to study algebras abstracted from their signature. The aim of this paper is to give a brief introduction to the theory thereof. We give basic definitions and examples, and we present several results and open problems, selected from almost one hundred years of ongoing research. We also discuss what is arguably the most important tool to study clones - the Galois connection between operations and relations built on the notion of preservation. We conclude the paper by explaining the connection between clones and the closely related category theoretic notion of Lawvere theory.

The Power of Peircean Algebraic Logic (PAL)

The existential graphs devised by Charles S. Peirce can be understood as an approach to represent and to work with relational structures long before the manifestation of relational algebras as known today in modern mathematics. Robert Burch proposed in [Bur91] an algebraization of the existential graphs and called it the Peircean Algebraic Logic (PAL). In this paper, we show that the expressive power of PAL is equivalent to the expressive power of Krasner-algebras (which extend relational algebras). Therefore, from the mathematical point of view these graphs can be considered as a two-dimensional representation language for first-order formulas. Furthermore, we investigate the special properties of the teridentity in this framework and Peirce’s thesis, that to build all relations out of smaller ones we need at least a relation of arity three (for instance, the teridentity itself).

The Lattice of Compatible Quasiorders of Acyclic Monounary Algebras

Acyclic monounary algebras are characterized by the property that any compatible partial order r can be extended to a compatible linear order. In the case of rooted monounary algebras

IRREDUCIBLE QUASIORDERS OF MONOUNARY ALGEBRAS

{\cal A}=(A,f)

The teridentity and peircean algebraic logic

we characterize the intersection of compatible linear extensions of r by several equivalent conditions and generalize these results to compatible quasiorders of