abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that "division" is always possible. Quasigroups differ from groups...

generalized to Moufang loops by Stephen Gagola III in recent years. Any quasigroup satisfying one of the Moufang identities must, in fact, have an identity...

In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of...

with quasigroups, which are generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve...

2004, as a part of his PhD dissertation entitled Totally Antisymmetric Quasigroups. The Damm algorithm is similar to the Verhoeff algorithm. It too will...

Each quasigroup is isotopic to a loop. Let ( Q , ⋅ ) {\displaystyle (Q,\cdot )} and ( P , ∘ ) {\displaystyle (P,\circ )} be quasigroups. A quasigroup homotopy...

prove that A forms a medial quasigroup under this operation. The Bruck–Murdoch-Toyoda theorem states that every medial quasigroup is of this form, i.e. is...

n,R]=[R,R,n]=\{0\}\ .} The nucleus is an associative subring of R. A quasigroup Q is a set with a binary operation ⋅ : Q × Q → Q {\displaystyle \cdot...

a CH-quasigroup, introduced by Manin (1986, definition 1.3), is a symmetric quasigroup in which any three elements generate an abelian quasigroup. "CH"...

Identity (equation) Identity function Inverse element Monoid Pseudo-ring Quasigroup Unital (disambiguation) Weisstein, Eric W. "Identity Element". mathworld...

Latin squares and finite quasigroups are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic...

Commonly studied types of magma include: Quasigroup: A magma where division is always possible. Loop: A quasigroup with an identity element. Semigroup: A...

a−1 ∗ (a ∗ c), which implies b = c by associativity. For example, every quasigroup, and thus every group, is cancellative. To say that an element a in a...

unique for all a and all b (the Latin square property) is a quasigroup. In a quasigroup, division in this sense is always possible, even without an identity...

unique solutions and we get an n-ary quasigroup. An n-ary group is an n-ary semigroup which is also an n-ary quasigroup. Post gave a structure theorem for...

Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Rng...

512 bits. It uses quasigroup string transformations with quasigroups of order 264, defined by extended Feistel networks. The quasigroups used in every iteration...

differential equations, rather than out of abstract algebra (such as quasigroup, for example). The modern theory of pseudogroups was developed by Élie...

Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Rng...

Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Rng...

then the rows are orthogonal. In fact, small category is orthogonal to quasigroup, and groupoid is orthogonal to magma. Consequently there are zeros in...

commutative. Quasigroup: a magma obeying the Latin square property. A quasigroup may also be represented using three binary operations. Loop: a quasigroup with...

Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Rng...

idempotent, commutative quasigroup. It has the additional property that ab = c implies bc = a and ca = b. Conversely, any (finite) quasigroup with these properties...

of X. Free loop Loop group Loop space Loop algebra Fundamental group Quasigroup Adams, John Frank (1978), Infinite Loop Spaces, Annals of mathematics...

Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Rng...

Steiner triple system. As such, it can be given the structure of a quasigroup. This quasigroup coincides with the multiplicative structure defined by the unit...

quasigroup, IP-quasigroup, F-quasigroups, CI-quasigroup, I-quasigroups Bol loops, totally symmetric quasigroups and quasigroup Stein et al.) Quasigroups a balanced...

However, Cayley tables can also be used to characterize the operation of a quasigroup, which does not assume associativity as an axiom (indeed, Cayley tables...

Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Rng...