Algebraic Structures Simpler than Groups

GROUPOIDS

A function f is called a binary operation (or a composition mapping) on a set S if f: S × S → S.

A groupoid is an ordered pair (S, ·) where · is a binary operation on S. The set S is called the underlying set of the groupoid and the function · is called the groupoid operation. We say that S is a groupoid under f. A groupoid is also called a binary algebra. The order of a groupoid (S, ·) is card(S).

If S is a finite set, {a_{1}, a_{2}, …, a_{n}}, the Cayley table (or composition table) for the groupoid (S, ·) is an n×n rectangular table whose (i, j) element is a_{i}a_{j}.

Let (S, ·) be a groupoid.

The operator · is if
commutative  ab = ba for all a, b ∈ S
associative  (ab)c = a(bc) for all a, b, c ∈ S

Let A and B be groupoids and let h be a function such that h(xy) = h(x)h(y) for all x, y ∈ A.

If  then h is called
h: A ⟶ B a homomorphism
h: A ⟶^{onto} B an epimorphism
h: A ⟶^{1-1} B a monomorphism
h: A ⟶_{onto}^{1-1} B an isomorphism
h: A ⟶ A an endomorphism
h: A ⟶_{onto}^{1-1} A an automorphism

Groupoids A and B are isomorphic if there is an isomorphism h: A → B.

Let (S, ·) be a groupoid and A ⊆ S.

Then A is  if
closed under · AA ⊆ A
a left ideal  SA ⊆ A
a right ideal AS ⊆ A
an ideal  SA ⊆ A and AS ⊆ A

An ideal is sometimes called a two-sided ideal.

Let x ∈ S and A ⊆ S.

Then is called a
Sx  a principal left ideal
xS  a principal right ideal

TYPES OF GROUPOIDS

Let (S, ·) be a groupoid.

The groupoid is if
commutative  · is commutative
Abelian  · is commutative
a semigroup  · is associative

Degenerate groupoids

A groupoid S is called if
null  card(S) = 0
trivial  card(S) = 1

SPECIAL GROUPOID ELEMENTS

Let (S, ·) be a groupoid and a ∈ S.

The element a is called if for all x, y ∈ S
a left identity  ax = x
a right identity  xa = x
an identity  ax = x and xa = x
left zero  ax = a
right zero  xa = a
zero  ax = a and xa = a
left cancellable  ax = ay ⇒ x = y
right cancellable  xa = ya ⇒ x = y
cancellable  {ax = ay ⇒ x = y and \atop xa = ya ⇒ x = y}
an idempotent  a^{2} = a

An identity is sometimes called a two-sided identity. A left zero is said to be left absorbent and a right zero is said to be right absorbent. A zero element is said to be absorbent, or two-sided absorbent.

Let 0 ∈ S be a zero element. Let x ∈ S and A ⊆ S.

Then  if
x annihilates A on the left xA = {0}
x annihilates A on the right Ax = {0}
x is nilpotent  x^{n} = 0 for some n ∈ ℕ
A is nilpotent  A^{n} = {0} for some n ∈ ℕ

HALFGROUPOIDS

A halfgroupoid is an ordered pair (S, f) where f is a function such that f: H → S for some H ⊆ S. The order of S is card(S). If H ⊆ S, then H is a subhalfgroupoid if H is a halfgroupoid such that ab = c in S whenever ab = c in H.

Let H be a subhalfgroupoid of G

We say that  if
H is closed in G  {a, b ∈ H, ab = c in G ⇒ \atop ab = c in H}

G is an extension of H (i) ab = c in G ⇒ a, b ∈ H

(ii) c ∈ G, c ∉ H ⇒ c = ab
in G for some a, b ∈ H

H generates G the only closed subhalfgroupoid of G containing H as a subhalfgroupoid is H

Let G be an extension of H.

We say that if
G is complete ab is defined in H for all a, b ∈ G

G is open (i) ab = c in G, c ∈ H ⇒ ab = c in H (ii) c ∈ G, c ∉ H, ab = a′b′ = c in G ⇒

a = a′, b = b′

SUBGROUPOIDS

Let (S, ·) be a groupoid and let A ⊆ S. If A is closed under ·, then A is called a subgroupoid of S.

If f: A → S and g: B → S are homomorphisms then we say g extends f to B if A is a subgroupoid of B and f(a) = h(a) for all a ∈ A. We also say g induces f on A.

Let ≡ be an equivalence relation on the set S. Then ≡ is called a congruence for · if a ≡ c and b ≡ d ⇒ ab ≡ cd. The equivalence classes are called congruence classes. The quotient sets are called quotients. The groupoid S/E under ·/ ≡ is called a quotient groupoid of S.

If A ⊆ S, we say that A is normal if A = ∅ or if there is a congruence E on S such that A is an E-class.

Let ∅ ≠ A = A_{0} ⊂ A_{1} ⊂ ⋯ ⊂ A_{m} = B be a finite chain of subgroupoids of S. Then this chain is called a normal series if A_{i−1} is normal in A_{i} for 1 ≤ i ≤ m.

If B is a normal series from A to B and contains A, then B is a refinement of A. The chain A is unrefinable if it has no proper refinement.

The subgroupoid A is a subnormal subgroupoid of B if there is a normal series from A to B.

SEMIGROUPS

A semigroup is an associative groupoid.

Types of semigroups a semigroup such that

A cancellation semigroup every element is cancellable
A monoid  with an identity

A cyclic semigroup every element is of the form

c^{n}

A Gaussian semigroup commutative cancellation semigroup S with identity such that every nonunit of S is a product of irreducible non-units in essentially one way

A GCD semigroup a semigroup S such that any two elements a, b ∈ S have a greatest common divisor

An LCM semigroup a semigroup S such that any two elements a, b ∈ S have a least common multiple

A PL-semigroup a semigroup S such that I_{S} ∪ {∅} is closed under intersection and under union of chains.

If F is a semigroup and ∅ ≠ B ⊆ F, then F is called a free semigroup on B if each mapping f from B into a semigroup G can be extended to a unique homomorphism h of F into G. In this case, B is called a basis of F.

Divisibility in a semigroup S

We say that  if for some a ∈ S
x divides y on the right y = ax
x divides y on the left y = xa

Divisibility in a commutative semigroup S

We say that  if
x|y  y = ax for some a ∈ S
x divides y  x|y
y is a multiple of x  x|y
y is a left associate of x Sx = Sy
y is a right associate of x xS = yS
x is a unit  x is invertible

d is a common divisor of x and y

d|x and d|y

m is a common multiple of x and y

x|m and y|m

d is a greatest common divisor of x and y

{d|x and d|y and \atop d^{∗}|x, d^{∗}y ⇒ d^{∗}|d}

m is a least common multiple of x and y

{x|m and y|m and \atop x|m^{∗}, y|m^{∗} ⇒ m|m^{∗}}

x is a divisor of y  x|y
x if a factor of y  x|y

Primes in a commutative semigroup S

We say that x is if for all a, b ∈ S
irreducible  x = ab ⇒ x|a or x|b
prime  x|ab ⇒ x|a or x|b

MONOIDS

A monoid is a semigroup with an identity.

A cyclic monoid is a monoid that consists of the powers c^{n} of some element c. It is said to be generated by c.

QUASIGROUPS

If F is a monoid and ∅ ≠ B ⊆ F, then F is called a free monoid on B if each mapping f from B into a monoid G can be extended to a unique homomorphism h of F into G. B is called a basis of F.

A groupoid S is called a quasigroup if any two of a, b, c ∈ S in ab = c uniquely determines the third.

An inversive quasigroup is one in which a, b ∈ S implies there is a unique d ∈ S such that da = b and for any c ∈ S, there is a c′ ∈ S such that for all a ∈ S, (ac)c′ = a.

The nucleus of an inversive quasigroup is {x · x′|x ∈ S}.

LOOPS

A loop is a quasigroup with an identity.

A totally symmetric loop is a commutative loop that satisifes a(ab) = b for all a, b ∈ S.

GROUPS

A group is a monoid in which every element is invertible.

There are many equivalent definitions and many types of groups, but this survey is meant only to discuss structures simpler than a group.