Bidirectional named sets as structural models of interpersonal communication†

Treating communication as information exchange between systems, we employ the most fundamental structure in mathematics, nature and cognition, which is called a named set or a fundamental triad because it has been useful in a variety of areas such as networks and networking, physics, information theory, mathematics, logic, database theory and practice, artificial intelligence, mathematical linguistics, epistemology and methodology of science, to mention but a few. Here we use structural models based on the theory of named sets for description and analysis of interpersonal communication explicating its structural regularities.


Named sets and fundamental triads
We consider three primary types of named sets and fundamental triads [4]. A basic fundamental triad or a basic named set has the following form (1).
It is a triad X = (X, f, I), in which X and N are two objects and f is a correspondence (e.g., a binary relation) between X and I. With respect to X, X is called the support of X, N is called the component of names (reflector) or set of names of X, and f is called the naming correspondence (reflection) of X. Note that here, f is not necessarily a mapping or a function.
The standard example is a basic named set (fundamental triad), in which X consists of people, N consists of their names and f is the correspondence between people and their names. Another example is a basic named set (fundamental triad), in which X consists of things, N consists of their names and f is the correspondence between things and their names [7]. It is necessary to make a distinction between triples and triads. A triple is any set with three elements, while a triad is a system of three connected elements (components). It is worthy of note that mathematicians introduced the concept of a triple in an abstract category [8]. In essence, such a triple is a triad that consists of three fundamental triads and thus is a triad of the second order [4].
Understanding of the complex nature of the categorical triple made mathematicians to change the name of this structure and now it is always called a monad [9]. Interestingly, this shows connection between fundamental triads and Leibniz monads. A bidirectional fundamental triad or a bidirectional named set has the following form (2).
It is also a triad D = (X, f, Y), in which the naming relation f goes in two directions.
We have an example of a bidirectional named set when two people are exchanging messages, e.g., be e-mails, messaging or talking to one to another. In this case, X and Z are people while f and g are messages that go from one person to another. Note that when mathematicians or computer scientists use connections without direction such as those that are used, for example, in general graphs [10], these connections actually have both directions and are more explicitly represented by the union of directed connections h and g. A cyclic fundamental triad or a cyclic named set has the following form The following graphic form (4) can also describe it. X (4) f An example of a cyclic named set is a subatomic particle, such as an electron, which acts on itself (cf., for example, [11]). Another example of a cyclic named set is a computer network. In it, X consists of computers and f contains all connections between them. Let us obtain some simple properties of named sets related to their compositions. Proposition 1. a) The sequential composition of basic named sets is a basic named set. b) The sequential composition of bidirectional named sets is a bidirectional named set. c) The sequential composition of cyclic named sets is a cyclic named set. In many cases, a bidirectional named set can be decomposed into the inverse composition of two basic named sets as it is demonstrated in the following diagram, which is a decomposition of Diagram (2). with the naming correspondence (a binary relation in this case) This shows how it is possible to construct bidirectional named sets using inverse composition of basic named sets [4]. Inverse composition of basic named sets X = (X, f, I) and Y = (Y, g, J) is defined as Proposition 3. The inverse composition of named sets X and Y is equal to the sequential composition of X and the involution Y o of Y, i.e., X ≬ Y = X ∘ Y o Although any bidirectional named set is the inverse composition of basic named sets, it is a fundamental structure such as a set, graph, category, fuzzy set or multiset. In more detail, relations between basic and bidirectional named sets are studied elsewhere. There are also other compositions of named sets [4]. If X = (X, r, I ) and Y = (Y, q, J ) are named sets, then their sequential composition X ∘ Y is the named set (X, roqo , J ) where ro = r  (X  (I  Y)) and qo = q  ((I  Y)  J). Example 1. Superposition of functions is the sequential composition of the corresponding named sets in the case when I = Y. Example 2. Composition of morphisms in categories is the sequential composition of the corresponding named sets. If X = (X, r, I ) and Y = (Y, q, J ) are named sets, then their parallel composition X  Y is An important special case of inverse composition is cyclic composition of named sets. If X = (X, r, Y) and Y = (Y, q, X ) are two named sets, in which the support of X coincides with the reflector of Y and the support of Y coincides with the reflector of X, then their cyclic composition has the form X  Y = (X, r, Y)  (Y, q, X) = (X, r∘q, X) If X = (X, r, Y) and Y = (Y, q, Z) are two named sets, then their chain composition is a named set chain (Burgin, 2011) and has the form V = [ X, Y ]

Interpersonal communication
People understand communication either as a process of information exchange or as a result of such a process. In addition, communication can include exchange of ideas, thoughts and/or opinions. However, everything that is transmitted in communication comes through information exchange. As a result, it is natural to treat communication as a system of information transmissions, which can be organized in a sequence or go concurrently. The action of information transmission has the structure of a basic named set (6), in which its support and reflector have the roles of a sender and receiver. t Communication as a pure exchange of information in the form of messages has the structure (7) of a decomposed bidirectional named set, in which the naming relation represents messaging. According to the general theory of information, information for a system R is a capacity to change an infological system IF(R) of the system R. There are three basic forms of information organization:  Quantization by determining units of information and then measuring or counting these units  Qualification, in which information is represented in an explicit form pertinent to the problem or situation, e.g., by modeling or describing  Categorization, e.g., classification or clustering All considered above types and schemas represented direct communication. Mediated communication has the structure of a chain (10)