We prove that tree isomorphism is not expressible in the language FO TC COUNT This is surprising since in the presence of ordering the language captures NL whereas tree isomorphism and canonization are in L Lin Our proof uses an Ehrenfeucht Fra ss e game for transitive closure logic with counting Gr a IL As a corresponding upper bound we show that tree canonization is expressible in FO COUNT lo...

Following Richter (1966), we provide criteria under which a preference relation implied by a finite set of choice observations has a complete extension that can in turn be represented by a utility function. These criteria rely on a mapping over preference relations, the rational closure, which is a generalization of the transitive closure and is employed to construct the complete extension. We ...

In this paper, we introduce intuitionistic fuzzy incline (IFI), intuitionistic fuzzy incline matrix (IFIM) and its determinant. Also the transitive closure, power of convergent, nilpotence of IFIM and adjoint of an IFIM are considered here. Some properties of determinant of IFIM and triangular IFIM are also introduced here. 2010 AMS Classification: 03E72, 15A15, 15B15

The concept of a closure operator ∇ in an ADL R was introduced. If ∇R is the set of all ∇−invariant elements of R, then the concepts of ∇R−ideal, ∇R−prime ideal are introduced. The interrelations between ∇R−prime ideal and minimal prime ideal of R are derived. If B is the Birkhoff centre of R, then a sufficient condition is derived for a B−ideal to be a minimal prime ideal of R. Mathematics Sub...

In graphical or diagrammatic representations, not only a basic component of a diagram, but also a collection of multiple components can form a semantic unit, and it often helps reasoning with that diagram. For example, a row and a column in a table, or transitively closed nodes in a directed graph can be regarded as a semantic unit on its own. Designers have long noticed the importance of infor...

In this paper the description logic ALCNHR+(D) is introduced. Prominent language features beyond conjunction, full negation, and quantifiers are number restrictions, role hierarchies, transitively closed roles, generalized concept inclusions, and concrete domains. As in other languages based on concrete domains (e.g. ALC(D)) a so-called existential predicate restriction is provided. However, co...

Let P be a poset. A subset A of P is a k-family ii A contains no (k+1)-element chain. For i 1, let A i be the set of elements of A at depth i?1 in A. The k-families of P can be ordered by deening A B ii for all i, A i is included in the order ideal generated by B i. This paper examines minimal r-element k-families, deened as k-families A such that A = r and for every B < A, B < r. Minimal k-fam...

We build a theoretical framework that enables to extend the concept of transitive closure to the class of complete crisp relations, the class of reciprocal 3-valued relations and the class of reciprocal [0, 1]valued relations. We present algorithms to compute the transitive closure of reciprocal [0, 1]-valued relations, where the type of transitivity is either weak stochastic transitivity or st...

Let M and N be two representations of an extended Dynkin quiver such that the orbit O N of N is contained in the orbit closure O M and has codimension two. We show that the pointed variety (O M , N) is smoothly equivalent to a simple surface singularity of type A n , or to the cone over a rational normal curve.