Proceedings Bidirectional Named Sets as Structural Models of Interpersonal Communication

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1 Proceedings Bidirectional Named Sets as Structural Models o Interpersonal Communication Mark Burgin Department o Computer Science, University o Caliornia, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA; mburgin@math.ucla.edu Presented at the IS4SI 2017 Summit DIGITALISATION FOR A SUSTAINABLE SOCIETY, Gothenburg, Sweden, June Published: 8 June 2017 Abstract: Treating communication as inormation exchange between systems, we employ the most undamental structure in mathematics, nature and cognition, which is called a named set or a undamental triad because it has been useul in a variety o areas such as networks and networking, physics, inormation theory, mathematics, logic, database theory and practice, artiicial intelligence, mathematical linguistics, epistemology and methodology o science, to mention but a ew. Here we use structural models based on the theory o named sets or description and analysis o interpersonal communication explicating its structural regularities. Keywords: inormation; structure; structural inormation; interaction; correctness; knowledge 1. Introduction There are dierent models o communication (c., or example, [1 3]). They distinguish two basic types o communication interpersonal communication and intrapersonal communication. Interpersonal communication is inormation exchange between dierent systems. For instance, human-computer communication is interpersonal. Intrapersonal communication is inormation exchange in one system. For instance, communication in the World Wide Web is intrapersonal with respect to the World Wide Web. Here we use structural models based on the theory o named sets [4] or description and analysis o interpersonal communication explicating its structural regularities. Being the most undamental structure in mathematics, nature and cognition (as it is demonstrated in [4 6], named sets (also called undamental triads) have been useul in a variety o areas such as networks and networking, physics, inormation theory, mathematics, logic, database theory and practice, artiicial intelligence, mathematical linguistics, epistemology and methodology o science, to mention but a ew. Application o named set theory to communication theory is based on the act that communication as an inormation exchange has the intrinsic structure o a named set. In the next section, we consider elements o named set theory necessary or communication studies, which are exposed in Section Named Sets and Fundamental Triads We consider three primary types o named sets and undamental triads [4]. A basic undamental triad or a basic named set has the ollowing orm (1). N (1) It is a triad = (,, I), in which and N are two objects and is a correspondence (e.g., a binary relation) between and I. With respect to, is called the support o, N is called the component o Proceedings 2017, 1, 58; doi: /is4si

2 Proceedings 2017, 1, 58 2 o 6 names (relector) or set o names o, and is called the naming correspondence (relection) o. Note that here, is not necessarily a mapping or a unction. The standard example is a basic named set (undamental triad), in which consists o people, N consists o their names and is the correspondence between people and their names. Another example is a basic named set (undamental triad), in which consists o things, N consists o their names and 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 o three connected elements (components). It is worthy o note that mathematicians introduced the concept o a triple in an abstract category [8]. In essence, such a triple is a triad that consists o three undamental triads and thus is a triad o the second order [4]. Understanding o the complex nature o the categorical triple made mathematicians to change the name o this structure and now it is always called a monad [9]. Interestingly, this shows connection between undamental triads and Leibniz monads. A bidirectional undamental triad or a bidirectional named set has the ollowing orm (2). (2) N It is also a triad D = (,, Y), in which the naming relation goes in two directions. We have an example o a bidirectional named set when two people are exchanging messages, e.g., be s, messaging or talking to one to another. In this case, and Z are people while and g are messages that go rom one person to another. Note that when mathematicians or computer scientists use connections without direction such as those that are used, or example, in general graphs [10], these connections actually have both directions and are more explicitly represented by the union o directed connections h and g. A cyclic undamental triad or a cyclic named set has the ollowing orm (3) (3) The ollowing graphic orm (4) can also describe it. (4) An example o a cyclic named set is a subatomic particle, such as an electron, which acts on itsel (c., or example, [11]). Another example o a cyclic named set is a computer network. In it, consists o computers and contains all connections between them. Let us obtain some simple properties o named sets related to their compositions. Proposition 1. (a) The sequential composition o basic named sets is a basic named set. (b) The sequential composition o bidirectional named sets is a bidirectional named set. (c) The sequential composition o cyclic named sets is a cyclic named set. In many cases, a bidirectional named set can be decomposed into the inverse composition o two basic named sets as it is demonstrated in the ollowing diagram, which is a decomposition o Diagram (2). h Z (5) g Here = [h, g] or = h g. Thus, a decomposed bidirectional named set D is denoted by D = (, [h, g], Z) and has two components, which are basic named sets: The direct component = (,, Z) The inverse component Y = (Z, g, )

3 Proceedings 2017, 1, 58 3 o 6 Proposition 2. A bidirectional named set D is equal to the inverse composition o its components and Y. For instance, the set-theoretical bidirectional named set = ( = {1, 2, 3},, Z = {a, b}) with the naming correspondence (a binary relation in this case) = {(1, a), (3, b), (2, a), (b, 1), (a, 3)} is decomposable the inverse composition o the direct component Z = (, h, Z) and inverse component Y = (Z, g, ) where and h = {(1, a), (3, b), (2, a)} Y g = {(b, 1), (a, 3)} Y We see that in this case, = h g. This shows how it is possible to construct bidirectional named sets using inverse composition o basic named sets [4]. Inverse composition o basic named sets = (,, I) and Y = (Y, g, J) is deined as Y = (,, I) (Y, g, J) = ( J, g 1, I Y) Proposition 3. The inverse composition o named sets and Y is equal to the sequential composition o and the involution Y o o Y, i.e., Y = Y o Although any bidirectional named set is the inverse composition o basic named sets, it is a undamental structure such as a set, graph, category, uzzy set or multiset. In more detail, relations between basic and bidirectional named sets are studied elsewhere. There are also other compositions o named sets [4]. I = (, r, I) and Y = (Y, q, J) are named sets, then their sequential composition Y is the named set (, roqo, J) where ro = r ( (I Y)) and qo = q ((I Y) J). Example 1. Superposition o unctions is the sequential composition o the corresponding named sets in the case when I = Y. Example 2. Composition o morphisms in categories is the sequential composition o the corresponding named sets. I = (, r, I) and Y = (Y, q, J) are named sets, then their parallel composition Y is Y = (,, I) (Y, g, J) = ( J, g, I Y) An important special case o inverse composition is cyclic composition o named sets. I = (, r, Y) and Y = (Y, q, ) are two named sets, in which the support o coincides with the relector o Y and the support o Y coincides with the relector o, then their cyclic composition has the orm Y = (, r, Y) (Y, q, ) = (, r q, ) I = (, 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 orm V = [, Y]

4 Proceedings 2017, 1, 58 4 o 6 3. Interpersonal Communication People understand communication either as a process o inormation exchange or as a result o such a process. In addition, communication can include exchange o ideas, thoughts and/or opinions. However, everything that is transmitted in communication comes through inormation exchange. As a result, it is natural to treat communication as a system o inormation transmissions, which can be organized in a sequence or go concurrently. The action o inormation transmission has the structure o a basic named set (6), in which its support and relector have the roles o a sender and receiver. t (6) Sender (Source) o o Receiver (Sink) Communication as a pure exchange o inormation in the orm o messages has the structure (7) o a decomposed bidirectional named set, in which the naming relation represents messaging. Sender/Receiver o o Sender/Receiver (7) g Here each participant acquires the two-olded role o a Sender/Receiver. Note that both the Sender and Receiver can be not only individuals but also groups o individuals, devices, e.g., computers, birds, animals and other living beings. Connections t, and g between the Sender and the Receiver have three components: Communication space, e.g., a channel or a system o channels, is the medium, in which communication goes and which allows sending messages rom the sender to the receiver A system o messages, e.g., one message, where a message is the object sent (transmitted) A context consists o conditions (environment) in which communication goes Messages are carriers o inormation and usually have three components: The physical component o a message, e.g., electrical signals or piece o paper with some text. The structural component o a message, e.g., text or picture The mental component o a message, e.g., the meaning o a text Contexts usually have one o the ollowing three types: Individual contexts Group contexts General contexts A context determines how inormation is transmitted and also have three components: The physical component o a context, e.g., conditions, in which inormation is sent and received The structural component o a context, e.g., language o the message The mental component o a context, e.g., knowledge o the sender and receiver Besides, there is such a phenomenon as sel-communication. Sel-communication is communication sel-directed in the elaboration and sending o the message, sel-selected in the reception o the message, and sel-deined in terms o the ormation o the communication space. Sel-communication is special case o intrapersonal communication. Sel-communication is represented by the ollowing cyclic named set (8), in which is both the Sender and the Receiver. (8) Another important case o intrapersonal communication represented by a cyclic named set is communication in networks, such as the Internet, where each node can be both a receiver and sender. This process is also naturally modeled by a cyclic named set, in which is the whole network. A combination o network communication and sel-communication is called mass sel-communication.

5 Proceedings 2017, 1, 58 5 o 6 Enhanced communication as an exchange o inormation with inormation processing, which includes inormation organization, is the sequential composition (9) o two cyclic and one bidirectional named sets. Sender/Receiver o o Sender/Receiver g (9) p q According to the general theory o inormation, inormation or a system R is a capacity to change an inological system IF(R) o the system R. There are three basic orms o inormation organization: Quantization by determining units o inormation and then measuring or counting these units Qualiication, in which inormation is represented in an explicit orm pertinent to the problem or situation, e.g., by modeling or describing Categorization, e.g., classiication or clustering All considered above types and schemas represented direct communication. Mediated communication has the structure o a chain (10) o bidirectional named sets (undamental triads): o o o o. o o o (10) In a sequential communication, all messages are linearly ordered in time and transmission o each o them does not intersect in time with transmission o another one. There are dierent types o messages in communication: The starting message m0 is the irst message in communication. The concluding message m is the last message in communication. A eedback message is an outcoming message caused by some incoming message. A direct eedback message is an outcoming message caused by the previous incoming message. An initializing message is a message that is not a eedback message. Messages allow construction o characteristic structures o communication. Communication thread L consists o an initializing message called the root o L and a sequence o direct eedback messages, in which the irst element is the eedback to the initializing message and each next element is the direct eedback to the previous element. Communication leaping thread H consists o an initializing message called the root o H and a sequence o direct eedback messages, in which the irst element is the eedback to the initializing message and each next element is the eedback to the previous element but not necessarily direct eedback. Note that a communication thread is also a communication leaping thread. Communication multithread is the union o communication threads with a common root. Communication leaping multithread is the union o communication threads, which have a common root and can be leaping. Note that a communication thread is also a communication multithread and a communication multithread is also a communication leaping multithread. Communication hyperthread is the union o communication threads. Communication leaping hyperthread is the union o communication threads, some o which can be leaping. This gives us dierent types o communication. Communication is linear i it consists o a single communication thread. Communication is branching i it contains, at least, one communication multithread. The most general is concurrent communication. Two communication threads are disjoint i each o them does not have a eedback message to a message rom another one.

6 Proceedings 2017, 1, 58 6 o 6 Communication is disjoint i it consists o disjoint communication threads. Explication o structural peculiarities o communication is aimed at better organization o interpersonal communication in both human society and networks o artiicial devices such as cell phone networks or the Internet. Conlicts o Interest: The author declares no conlict o interest. Reerences 1. Harrah, D. Communication: A Logical Model; Cambridge University Press: Cambridge, UK, Keltner, J.W. Interpersonal Speech-Communication: Elements and Structures; Wadsworth P.C.: Belmont, CA, USA, Burgin, M.; Neishtadt, L. Communication and Discourse in Teachers Proessional Activity; Daugavpils Pedagogical Institute: Daugavpils, Latvia, Burgin, M. Theory o Named Sets; Mathematics Research Developments, Nova Science: New York, NY, USA, Burgin M. Theory o Named Sets as a Foundational Basis or Mathematics. In Structures in Mathematical Theories; D. Reidel Publishing Company: San Sebastian, Spain, 1990; pp Available online: 6. Burgin, M. Named Sets as a Basic Tool in Epistemology. Epistemologia 1995, VIII, Dalla Chiara, M.L.; di Francia, T.G. Individuals, kinds and names in physics. In Bridging the Gap: Philosophy, Mathematics, Physics; Kluwer: Dordrecht, The Netherlands, 1993; pp Barr, M.; Wells, C. Toposes, Triples, and Theories; Grundlehren der Math. Wissenschaten, v. 278; Springer: Berlin, Germany, Wadler, P. Comprehending monads. Math. Struct. Comput. Sci. 1992, 2, Berge, C. Graphs and Hypergraph; North Holland P.C.: Amsterdam, The Netherlands; New York, NY, USA, Close, F. The Ininity Puzzle: Quantum Field Theory and the Hunt or an Orderly Universe; Basic Books: New York, NY, USA, by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions o the Creative Commons Attribution (CC BY) license (