Multigroups (Q5903802)

An involutive semigroup of binary relations is any set of binary relations on some underlying set A closed under the binary operation of relative multiplication and the unary operation of inversion (or conversion) of binary relations. If all binary relations considered satisfy two conditions: (1) they are everywhere defined (that is, the domain of each of them coincides with A) and (2) they are single-valued (that is, each element of A cannot have two different images under any of these relations), then any involutive semigroup of relations becomes a group. By the celebrated Cayley representation theorem, the converse holds: every group is isomorphic to a group of permutations of a set. If we skip Condition (1) but preserve (2), we obtain involuted semigroups of one-to-one partial transformations. Each of them is an inverse semigroup, and conversely, every inverse semigroup is isomorphic to an involuted semigroup of one- to-one partial transformations (the Wagner-Preston representation theorem). The paper considers the situation when Condition (1) is preserved, while Condition (2) is omitted. In such a situation we arrive at involuted semigroups of relations which map all of A onto all of A in a not necessarily single-valued way. Such relations are called ``multipermutations'', and involuted semigroups isomorphically representable by multipermutations are called ``multigroups''. It turns out that multigroups do not form a variety of algebras. They do form a quasi variety, the corresponding quasi identities (also called conditional identities) are instances of a simple scheme of axioms. They are infinite in number, and no finite set of elementary axioms can characterize the class of multigroups. These results lead us to the heart of the paper, which are the so-called multigroups with inverting involution (i.e., such, that \(xx^{-1}x=x\) for every element x, which means that \(x^{-1}\) is an inverse for every x). A binary relation satisfying this condition is precisely a difunctional relation in the sense of J. Riguet, and we arrive at involuted semigroups of difunctional multipermutations. It turns out that many familiar semigroups of multi-valued transformations belong to this type. For example, if V is any vector space, one may consider the set M(V) of all multi-valued linear mappings of V onto V. Then M(V) is an involuted semigroup of difunctional multipermutations. It turns out that every involuted semigroup of difunctional multipermutations is an inverse semigroup and conversely, every inverse semigroup is isomorphic to an involuted semigroup of difunctional multipermutations. This is an entirely new representation theorem for inverse semigroups, alternative to that of Wagner-Preston. The author argues why this type of representations may be interesting and observes that an entirely new type of results on inverse semigroups and a new approach to the theory of inverse semigroups becomes possible because of this representation theorem.