Abstract
The purpose of this long paper was to develop the theories of generalised heaps and generalised groups in their mutual connections. To this end, Wagner began by introducing the new notion of a semiheap: a system with a ternary operation satisfying certain conditions. He explored some of the basic properties of semiheaps, as well as setting out elements of the theory of binary relations, the use of which was central to his approach. He next moved to the consideration of semigroups with involution, which turn out to have a natural connection with semiheaps, namely that any semiheap may be embedded in such a semigroup. Wagner then restricted his attention to a specific class of semigroups with involution: generalised groups (a.k.a. inverse semigroups), and the class of semiheaps with which they are closely associated: generalised heaps. He established elementary theories for these objects, and showed, for example, that any generalised heap may be embedded in a generalised group. These theories were then further expanded via the exploration of certain special binary relations in generalised heaps and generalised groups: the compatibility relation and the canonical order relation. The final section of the paper applies the previously developed notions to the context of binary relations and partial mappings and transformations: semiheaps and generalised heaps have a natural interpretation as abstractions of systems of binary relations or partial mappings between different sets, whilst semigroups and generalised groups apply in the case of partial transformations of a single set. It is proved that every generalised heap admits a representation by means of partial mappings, whilst every generalised group admits a representation via partial transformations.
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Notes
- 1.
In the theory of binary relations, we use the notation of Bourbaki [3]: \(\mathop{\rho }\limits^{-1}\) denotes the inverse of a binary relation ρ, and σ ∘ρ denotes the product of binary relations ρ and σ.
- 2.
For the case of commutative groups, it was considered earlier by Prüfer [10].
- 3.
- 4.
For the definition of the union of sets, see [3].
- 5.
Our use of our notation from mathematical logic is inessentially distinct from that of Lorenzen [15]. The advantage of the notation of Lorenzen over that of other authors is that, on the one hand, it best expresses the principle of duality, and, on the other, it accords with the corresponding notation in set theory. The similarity of the notation for the universal quantifier and conjunction, and of that for the existential quantifier and disjunction, is justified from the logical point of view by the fact that for the case of a finite domain [[550]] of variation of a variable, the universal quantifier is expressed by means of conjunction, and the existential quantifier by disjunction. Furthermore, this corresponds to the notation by similar signs of the intersection of two subsets or of an arbitrary set of subsets, and of the union of two subsets or of an arbitrary set of subsets.
- 6.
In contrast to Bourbaki [3], we will systematically employ the notion of multivalued functions in connection with the theory of binary relations.
- 7.
By \(\mathop{\rho }\limits^{n}\) we denote the n-th power of the binary relation ρ:
$$\displaystyle{\mathop{\rho }\limits^{n} =\mathop{\underbrace{ \rho \circ \cdots \circ \rho }}\limits _{n}.}$$ - 8.
This theorem is a generalisation of one proved by Lyapin [8] for semigroups.
- 9.
The empty subset of a semigroup may also be considered a subsemigroup.
- 10.
It is clear that the inverted semiheap of a generalised heap is a generalised heap.
- 11.
In the general case, if we have an arbitrary binary relation ρ ⊂ K × L, we can define a similar function in \(\mathfrak{P}(K)\), taking values in \(\mathfrak{P}(L)\), by means of the formula
$$\displaystyle{\rho (\mathfrak{k}) =\bigcap _{k\in \mathfrak{k}}\rho \langle k\rangle.}$$This function represents another form of the extension of the function defined by ρ to the set of subsets, which, in contrast to the union extension (8.1.26), is naturally called the intersection extension (see [12]).
- 12.
This theorem has been proved by Rees (see [11]) for the case of a generalised group of one-to-one partial transformations.
- 13.
Using the evident proposition that a generalised group considered as a generalised heap is a heap if and only if it is a group.
- 14.
By \(\overbrace{\varphi ^{-1}}^{-1}\), we mean the inverse binary relation for φ−1.
- 15.
From the definition (8.1.7) of lateral commutativity of a semiheap, it is clear that lateral commutativity is a necessary and sufficient condition for the identity transformation to be an anti-automorphism.
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Hollings, C.D., Lawson, M.V. (2017). Theory of Generalised Heaps and Generalised Groups. In: Wagner’s Theory of Generalised Heaps. Springer, Cham. https://doi.org/10.1007/978-3-319-63621-4_8
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