Abstract:
In this paper, we study the problem of element-by-element description of the set of all anti-endomorphisms of an arbitrary groupoid. In particular, the structure of the set of all anti-automorphisms of a groupoid is studied. It turned out that the set of all anti-endomorphisms of an arbitrary groupoid decomposes into a union of pairwise disjoint sets of transformations of a special form. These sets of transformations are referred to in this paper as basic sets of anti-endomorphisms. Each base set of anti-endomorphisms is parametrized by some mapping of the support set of the groupoid into a fixed set of two elements. These mappings are called the bipolar anti-endomorphism type. Since the base sets of anti-endomorphisms of various types have an empty intersection, each anti-endomorphism can uniquely be associated with its bipolar type. This assignment leads to a bipolar classification of anti-endomorphisms of an arbitrary groupoid. In this paper, we study the semiheap (3-groupoid of a special form) of all anti-endomorphisms. A subsemiheap of anti-endomorphisms of the first type and a subsemiheap of anti-endomorphisms of the second type are constructed. These monotypic semiheaps can be expressed into empty sets for specific groupoids. A conjecture is made about a subsemiheap of a special type of anti-endomorphisms of mixed type. The main research method in this work is the use of internal left and right translations of the groupoid (left and right translations). Since an arbitrary groupoid is considered, the set of all left translations (similarly to right translations) need not be closed with respect to the composition of transformations of the support set of the groupoid.
Keywords:groupoid endomorphism, anti-endomorphism, groupoid anti-automorphism, bipolar type of groupoid anti-endomorphism, groupoid, anti-endomorphism semiheap, monotypic anti-endomorphism semiheap.
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