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

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

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

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...

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

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

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...

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...

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

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

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 {\textstyle...

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...

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...

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

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

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

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...

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

containing it Microsoft Loop, collaborative software Loop (algebra), a quasigroup with an identity element Loop (graph theory), an edge that begins and...

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...

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

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

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...

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...

ambient space, which carries one subset to the other Isotopy of quasigroups. See Quasigroup#Homotopy and isotopy. Isotopy of loops, a triple of maps with...

Wiesław A. (2001), "On some old and new problems in n-ary groups" (PDF), Quasigroups and Related Systems, 8: 15–36, MR 1876783. Eliel, Ernest; Wilen, Samuel;...