Authors: Yakov A. Iosilevskii
There are two presently common onamastic (onomatological) methods of logographically naming and thus concisely describing an algebraic system; both methods are often used simultaneously. According to one method, an algebraic system is equivocally denoted by an atomic logographic symbol that originally denotes a certain underlying set of elements, which is regarded as the principal one, while all other objects of the algebraic system, properly named, are kept in mind and are regarded as implicit properties of that set or of its separate elements. That is to say, according to this method, an algebraic system is its principal underlying set of elements together with all its properties, which are implied and are not mentioned explicitly. According to the other method, an algebraic system is regarded as an ordered multiple, whose coordinates properly denote the defining objects of the algebraic system, and consequently the ordered multiple name is equivocally used as a proper name of the algebraic system. Thus, in this case, the togetherness of all constituents of the algebraic system is expressed by the pertinent ordered multiple name in terms of its coordinate names. In my recent article available at http://viXra.org/abs/1604.0124¸ I have demonstrated that both above onomastic methods are inconsistent. Therefore, in that article and also in my earlier article appearing at http://arxiv.org/abs/1510.00328, I suggested and used another onomastic method of logographically naming the pertinent algebraic systems, namely that employing, as a name of an algebraic system, a complex logographic name the union of all explicit constituent sets of the system, namely, the underlying sets of elements, the surjective binary composition functions, and the bijective singulary inversion functions; a function is a set (class) of ordered pairs. In the present article, the latter onomastic method is substantiated and generalized in two respects. First, the set of explicit constituent sets of an algebraic system is now extended to include the injective choice, or selection, functions of all additive and multiplicative identity elements of the algebraic system, belonging to its underlying sets, so that all those elements are now mentioned by the logographic name of the system. A general definition of an algebraic system is elaborated in such a way so as to make the new onomastic method universally applicable to any algebraic system.
Comments: 34 Pages. An additional category is "Mathematics: Set Theory and Logic"
[v1] 2016-05-02 05:34:51
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