Fuzzy rank functions in the set of all binary systems.

In this paper, we introduce fuzzy rank functions for groupoids, and we investigate their roles in the semigroup of binary systems by using the notions of right parallelisms and [Formula: see text]-shrinking groupoids.

Several types of groupoids induced by two-variable functions.

In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if [Formula: see text] is a groupoid and if [Formula: see text] is a function [Formula: see text], then [Formula: see text] is a left-twisted semigroup with respect to [Formula: see text] if for all [Formula: see text], [Formula: see text]. Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule [Formula: see text] for all [Formula: see text]. Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest is the class of groupoids defined over a field [Formula: see text] via a formula [Formula: see text], with [Formula: see text], fixed structure constants. Properties of these groupoids as twisted semigroups are discussed with several results of interest obtained, e.g., that in this setting simultaneous left-twistedness and right-twistedness of [Formula: see text] implies the fact that [Formula: see text] is a semigroup.

Age-dependent changes of stress and strain in the human heart valve and their relation with collagen remodeling.

In order to create tissue-engineered heart valves with long-term functionality, it is essential to fully understand collagen remodeling during neo-tissue formation. Collagen remodeling is thought to maintain mechanical tissue homeostasis. Yet, the driving factor of collagen remodeling remains unidentified. In this study, we determined the collagen architecture and the geometric and mechanical properties of human native semilunar heart valves of fetal to adult age using confocal microscopy, micro-indentation and inverse finite element analysis. The outcomes were used to predict age-dependent changes in stress and stretch in the heart valves via finite element modeling. The results indicated that the circumferential stresses are different between the aortic and pulmonary valve, and, moreover, that the stress increases considerably over time in the aortic valve. Strikingly, relatively small differences were found in stretch with time and between the aortic and pulmonary valve, particularly in the circumferential direction, which is the main determinant of the collagen fiber stretch. Therefore, we suggest that collagen remodeling in the human heart valve maintains a stretch-driven homeostasis. Next to these novel insights, the unique human data set created in this study provides valuable input for the development of numerical models of collagen remodeling and optimization of tissue engineering. STATEMENT OF SIGNIFICANCE: Annually, over 280,000 heart valve replacements are performed worldwide. Tissue engineering has the potential to provide valvular disease patients with living valve substitutes that can last a lifetime. Valve functionality is mainly determined by the collagen architecture. Hence, understanding collagen remodeling is crucial for creating tissue-engineered valves with long-term functionality. In this study, we determined the structural and material properties of human native heart valves of fetal to adult age to gain insight into the mechanical stimuli responsible for collagen remodeling. The age-dependent evolutionary changes in mechanical state of the native valve suggest that collagen remodeling in heart valves is a stretch-driven process.

The interaction between fuzzy subsets and groupoids.

We discuss properties of a class of real-valued functions on a set X (2) constructed as finite (real) linear combinations of functions denoted as [(X, ∗); µ ], where (X, ∗) is a groupoid (binary system) and µ is a fuzzy subset of X and where [(X, ∗); µ ](x, y): = µ (x ∗ y) - min{µ (x), µ (y)}. Many properties, for example, µ being a fuzzy subgroupoid of X, ∗), can be restated as some properties of [(X, ∗); µ]. Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin(X), □) for example.

(n - 1)-Step derivations on n-groupoids: the case n = 3.

We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping d is a self-derivation at each step. Following up on this idea we obtain several results and conclusions of interest. We also discuss the notion of a couplet (D, d) on X, consisting of a two-step derivation d and its square D = d ∘ d, for example, whose defining property leads to further observations on the underlying ranked trigroupoids also.

Fuzzy upper bounds in groupoids.

The notion of a fuzzy upper bound over a groupoid is introduced and some properties of it are investigated. We also define the notions of an either-or subset of a groupoid and a strong either-or subset of a groupoid and study some of their related properties. In particular, we consider fuzzy upper bounds in Bin(X), where Bin(X) is the collection of all groupoids. Finally, we define a fuzzy-d-subset of a groupoid and investigate some of its properties.

On abelian and related fuzzy subsets of groupoids.

We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated.