Richman, Fred

The context for the development of this work is constructive mathematics
without the axiom of countable choice. By constructive mathematics, we mean mathematics
done without the law of excluded middle. Our original goal was to give a list
of axioms for the real numbers R by only considering the order on R. We instead
develop a theory of ordered sets and their completions and a theory of ordered abelian
groups.

The earliest known system of formal logic, syllogistic logic , was set forth by Aristotle in Prior Analytics, which appeared around 350 B.C. A syllogism contains three statements, two premises and a conclusion. Each of the statements is one of four types, A, E, I, or O, which determines the mood of the syllogism. The four possible figures of a syllogism are determined by the arrangement of the terms within the statements. The form of a syllogism consists of both a mood and a figure, so there are 256 possible forms. A valid syllogism is one in which if the premises are true, then the conclusion must also be true. While Aristotle found 19 valid forms of the 256 possible, and others later found 24, it is now generally agreed that there are 15. Together, these consist of all possible ways to construct valid arguments within this system of syllogistic logic. We can use the rules of this system and certain operations to change statements and convert one valid form to another. We can also present these forms using various geometric representations.

The concept of valuated groups, simply presented groups, and simultaneous decomposition of an abelian group and a subgroup are discussed. We classify the structure of finite valuated p-groups of order up p^4. With a refinement of a classical theorem on bounded pure subgroups, we also relate the decomposition of a finite valuated p-group to the simultaneous decomposition of a finite abelian p-group and a subgroup.