Multilayer networks as embodied consciousness interactions. A formal model approach.

An algebraic interpretation of multilayer networks is introduced in relation to conscious experience, brain and body. The discussion is based on a network model for undirected multigraphs with coloured edges whose elements are time-evolving multilayers, representing complex experiential brain-body networks. These layers have the ability to merge by an associative binary operator, accounting for biological composition. As an extension, they can rotate in a formal analogy to how the activity inside layers would dynamically evolve. Under consciousness interpretation, we also studied a mathematical formulation of splitting layers, resulting in a formal analysis for the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal efficacy of conscious interactions, predicting topological network changes after conscious layer interactions. Our approach provides a mathematical account of coupling and splitting layers co-arising with more complex experiences. These concrete results may inspire the use of formal studies of conscious experience not only to describe it, but also to obtain new predictions and future applications of formal mathematical tools.

The logic of orthomodular posets of finite height

Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. A central question is how to introduce implication in such a logic. We give a positive answer whenever the orthomodular poset in question is of finite height. The crucial advantage of our solution is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and that its implication operator is everywhere defined. We present here a complete list of axioms for implication orthomodular posets. This enables us to derive an axiomatization in Gentzen style for the algebraizable logic of orthomodular posets of finite height.

Sublattices and Δ-blocks of orthomodular posets

States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$-blocks with respect to Boolean subalgebras and distributive subposets they include.

Optimization of the reducible objective functions with monotone factors subject to FRI constraints defined with continuous t-norms

In this paper, we investigate a special kind of optimization with fuzzy relational inequalities constraints where a continuous t-norm is considered as the fuzzy composition and the objective function can be expressed as in which and are increasing and decreasing functions, respectively, and is a commutative and monotone binary operator. Some basic properties have been extended a necessary and sufficient condition is presented to realize the feasibility of the problem. Also, an algorithm is given to optimize the objective function on the region of the FRI constraints. Finally, five examples are appended with two continuous t-norms, Lukasiewicz and Yager, and different objective functions, for illustrating.

Optimization of the reducible objective functions with monotone factors subject to FRI constraints defined with continuous t-norms

In this paper, we investigate a special kind of optimization with fuzzy relational inequalities constraints where a continuous t-norm is considered as the fuzzy composition and the objective function can be expressed as in which and are increasing and decreasing functions, respectively, and is a commutative and monotone binary operator. Some basic properties have been extended a necessary and sufficient condition is presented to realize the feasibility of the problem. Also, an algorithm is given to optimize the objective function on the region of the FRI constraints. Finally, five examples are appended with two continuous t-norms, Lukasiewicz and Yager, and different objective functions, for illustrating.

Fixed Point Theorems on Nonlinear Binary Operator Equations with Applications

The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone and partial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some corresponding results are obtained. Finally, the applications of our results are given.