Forms of a binary operator satisfying the constraints of a series of different axiom systems. The successive blocks of results in each case show forms of the operator allowed with 2, 3 and 4 possible elements. Note that with 3 and 4 elements, only forms inequivalent under interchange of element labels are shown. Representations of notable systems in mathematics are: (1) semigroup theory, (2) commutative group theory, (3) basic logic, (4) commutative semigroup theory, (5) squag theory, (6) group theory, (7) junctional calculus, (8) equivalential calculus and (9) implicational calculus. In each case the operator forms shown correspond to possible semigroups, commutative groups, systems of logic (Boolean algebras), etc. with 2, 3 and 4 possible elements. The operator forms shown can be thought of as giving multiplication tables. In model theory, these forms are usually called the models of an axiom system.
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