%path = "maths/stuctures/group" %kind = kinda["texts"] %level = 10
Group-like algebraic structures consist of a set \(M\) with a binary operation \(\circ\), i.e. \((M,\circ)\). For all elements (\(\forall_{a,b\in M}\)):
\(a\circ b \in M\) Closedness \(\rightarrow\) Magma
\(a\circ(b\circ c) = (a\circ b)\circ c\) Associative Law \(\rightarrow\) Semigroup
\(a^n = a\) Idempotent Semigroup \(\rightarrow\) Lattice
\(\exists_e|e\circ a = a\circ e = a\) Neutral Element \(\rightarrow\) Monoid
\(\exists_\bar{a}|\bar{a}\circ a = a\circ\bar{a} = e\) Inverse Element \(\rightarrow\) Group
\(a\circ b = b\circ a\) Commutative Law \(\rightarrow\) commutative or abelian Group