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A triple of bijections (γ;α,β)
from one binary system onto another is called an isotopism if the maps transfer the
multiplication: (xy)γ = xαyβ. These maps have been used by Albert, Bruck and
others in the study of quasigroups and loops. A natural generalization of isotopism is
given to a triple of maps called a topism, which transfers just the multiplication, as
above, from one binary system into another. It will be shown, (1.7), that a topism
from one quasigroup into another is an isotopism if and only if any one of the maps is
one-to-one and onto.
Two maps, α,β on a binary system (A,⋅) induce a new binary operation, ∘,
defined in the natural fashion x ∘ y = xαyβ. The relationship between topic
imbeddings and induced systems is studied in §2. It is shown, (2.3), that one
groupoid is topically imbeddable in another precisely when it is isomorphic to a
subgroupoid of an induced groupoid of the second. Thus, (2.7), two quasigroups
are isotopic if and only if one is isomorphic to an induced groupoid of the
other.
Finally, the imbeddings of nonassociative binary systems into semigroups and
groups are considered. It is shown, (3.1) and (3.2), that groupoids can be imbedded
as ideals in semigroups. However, it is seen (3.4) that a groupoid with identity is
topically imbeddable in a group precisely when the groupoid is isomorphic to a
subsemigroup, with identity, of the given group. From this a generalization, (3.6), of
Albert’s Theorem that a loop is isotopic to a group if and only if they are isomorphic
is obtained.
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