# IJPAM: Volume 98, No. 2 (2015)

**BOOLEAN ALGEBRAS AND CLASSIFICATION OF**

INTERACTIONS IN SUFFICIENT-COMPONENT

CAUSE MODEL

INTERACTIONS IN SUFFICIENT-COMPONENT

CAUSE MODEL

Institute of Industrial Ecology of Ural Branch of RAS

S. Kovalevskaya st., 20, Ekaterinburg, RUSSIA

Ural Federal University

V. Lenin pr., 51, Ekaterinburg, RUSSIA

**Abstract. **A mathematical model of the sufficient-component cause framework is considered based on the theories of Boolean algebra. The model consists of the space of states of a binary experiment and a set of symmetries of the experiment. The space of states is a Boolean algebra of n Boolean variables where n is the number of the binary causes in the experiment. The set of symmetries of the experiment is a subgroup of the group of all automorphisms of Boolean algebra of the states of experiment. This subgroup is generated by transformations preserving a type of interaction. An experimenter should deduce these transformations from the peculiar properties of the experiment. Examples of such transformations are provided. Classification of interactions is obtained by the calculation of the orbits of action of the group of symmetries on the space of states of the experiment. It is shown that the classification of the interaction for the ordinary symmetries of sufficient causes is the same as reported in related works. Other symmetries of the binary experiment are considered as well. It is shown that the corresponding classification of the interaction types in a binary experiment depends substantially on the symmetries of the experiment. Statistical criteria of particular types of responses are proven and the problem of mutual antagonism is discussed in the Appendix.

**Received: **October 1, 2014

**AMS Subject Classification: **03G05, 06E30, 94C10, 22F50

**Key Words and Phrases: **sufficient component cause model, Boolean automorphisms, group action, orbits, synergism, antagonism

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**DOI: 10.12732/ijpam.v98i2.7**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2015

**Volume:**98

**Issue:**2

**Pages:**239 - 259

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**This work is licensed under the Creative Commons Attribution International License (CC BY).**