An elementary logical system (ELS) is a commutative diagram that is characterized by the equation IE 1 = E 2I, where E 1 : X → X and E 2 : Y → Y are external operators and I : X → Y is an internal operator. The sets X and Y are thought of as vector spaces of “input objects” and “output objects,” respectively, that have the same dimensions; and I is thought of as an operator such that I ( X ) = Y. Theorem 1 establishes that if E 1 , E 2 , I ∈ U, where U is a non-necessarily associative Lie algebra, then E 2 ( Y ) = E 1 ( Y ) + [ I , E 1 ] ( X ), where [ , ] denotes Lie’s commutator, [ A , B ] = AB − BA. The general system logical theory (GSLT) is a theory of interconnections--for example, chains or hierarchies--of ELSs.
By means of Theorem 1, GSLT enables the language of commutative diagrams to be used for expressing the essential aspects of a surprising diversity of systems. These include (1) the dynamic state-transition behavior of many of the same systems that can be described via Resconi’s logical theory of systems [1]; (2) the three main theorems of Jessel’s theory of secondary sources in wave propagation and diffraction phenomena [2]; (3) certain mathematical relations that arise in the study of Riemann geometries; and (4) the key reactions in certain (non-linear) chemical control systems.
An important point--not mentioned by the authors--is that Theorem 1 is applicable only in systems where the term E 1 ( Y ) is well defined. In the particular example given for an instance of Resconi’s theory, it happens that this is not the case. However, the latter three systems are well defined, thus emphasizing the broad applicability of this elegant new theory.
