Partial algebras and their theories

The book starts with the estabishment of a collection of several more or less obvious algebraic properties (as axioms) of partial operations, expressed symbolically in a basic language. Any category satisfying these axioms is called dht-symmetric and as proposed by H.-J. Vogel in 2005 a Hoehnke category (or H-category). An example is the concrete category Par of sets and partial mappings. Then, by application of structural propositions on related fields of algebra, e. g. on partial orders and lattice theory, the second author could prove a completeness theorem for this type of categories, saying that each H-category is the subdirect product of H- subcategories of the concrete category Par. Consequently, the research could be continued in thas way, that the role of concrete partial operations could be replaced by abstract “arrows”. That means for the treatment of computational problems it suffices to implement the finite axioms of a H-category in the store of a computer and to use the fixed derivation rules which exist in a corresponding adequate programming language in order to obtain solutions of problems. Abstract H-categories are also called partial theories. Sometimes this treatment is also called metamathematics which is in use in Mathematical Logic. The next main steps (Chap. 5) are based upon an extended calculational work: Established is the first minimal system of generators and relations for a binary partial theory, and as a consequence it is obtained such a minimal system for the corresponding partial Dale monoid, while that given by E. C. Dale (even in the total case) was not minimal. Another result of the same extended calculations is the perception that H-categories are in 1-1 correspondence with the partial Dale monoids, yielding that former pairs are isomorphic iff the corresponding latter pairs are so (hence forming complete invariants of braids for H-categories). This result is quite fundamental and would be able to move categories from their traditionally central place in mathematics through the theory of braids of E. Artin. It lastly rests upon an idea of the famous B. H. Neumann to replace a free algebra and its clone of operations within a variety of universal algebras by braids from algebraic topology and would lead in future to further problems: to replace basic concepts by those of braid theory.