Showing papers by "Thomas Vetterlein published in 2019"

Orthogonality spaces of finite rank and the complex Hilbert spaces

Abstract: An orthogonality space is a set endowed with a symmetric, irreflexive binary relation. By means of the usual orthogonality relation, each anisotropic quadratic space gives rise to such a structure....

On the Coextension of Cut-Continuous Pomonoids

Abstract: We introduce cut-continuous pomonoids, which generalise residuated posets. The latter’s defining condition is that the monoidal product is residuated in each argument; we define cut-continuous pomonoids by requiring that the monoidal product is in each argument just cut-continuous. In the case of a total order, the condition of cut-continuity means that multiplication distributes over existing suprema. Morphisms between cut-continuous pomonoids can be chosen either in analogy with unital quantales or with residuated lattices. Under the assumption of commutativity and integrality, congruences are in the latter case induced by filters, in the same way as known for residuated lattices. We are interested in the construction of coextensions: given cut-continuous pomonoids K and C, we raise the question how we can determine the cut-continuous pomonoids L such that C is a filter of L and the quotient of L induced by C is isomorphic to K. In this context, we are in particular concerned with tensor products of modules over cut-continuous pomonoids. Using results of M. Erne and J. Picado on closure spaces, we show that such tensor products exist. An application is the construction of residuated structures related to fuzzy logics, in particular left-continuous t-norms.

Orthogonality spaces allowing gradual transitions

Abstract: We consider orthogonality spaces subject to the condition that gradual transitions between any two elements are possible. More precisely, given elements e and f , we require a homomorphism from the unit circle to the automorphism group to exist such that one of these automorphisms maps e to f , and any of these automorphisms leaves the elements orthogonal to e and f fixed. A natural example is the collection of one-dimensional subspaces of a real Hilbert space, endowed with the usual orthogonality relation. We show that the general case is closely related to this example.

The coextension of commutative pomonoids and its application to triangular norms

Abstract: Group coextensions of monoids, which generalise Schreier-type extensions of groups, have originally been defined by P.A. Grillet and J. Leech. The present paper deals with pomonoids, that is, monoids that are endowed with a compatible partial order. Following the lines of the unordered case, we define pogroup coextensions of pomonoids. We furthermore generalise the construction to the case that pomonoids instead of pogroups are used as the extending structures. The intended application lies in fuzzy logic, where triangular norms are those binary operations that are commonly used to interpret the conjunction. We present conditions under which the coextension of a finite totally ordered monoid leads to a triangular norm. Triangular norms of a certain type can therefore be classified on the basis of the presented results. Mathematics Subject Classification (2010): 06F05, 03B52. Keywords: Partially ordered monoid, coextension of pomonoids, triangular norm

Rees coextensions of finite tomonoids and free pomonoids

Abstract: A totally ordered monoid, or tomonoid for short, is a monoid endowed with a compatible total order. We reconsider in this paper the problem of describing the one-element Rees coextensions of a finite, negative tomonoid S, that is, those tomonoids that are by one element larger than S and whose Rees quotient by the poideal consisting of the two smallest elements is isomorphic to S. We show that any such coextension is a quotient of a pomonoid $$\mathcal R(S)$$ , called the free one-element Rees coextension of S. We investigate the structure of $$\mathcal R(S)$$ and describe the relevant congruences. We moreover introduce a finite family of finite quotients of $$\mathcal R(S)$$ from which the coextensions arise in a particularly simple way.