Codex @codex  0

A semigroup is an algebraic magma whose binary operation has the associative property. Its products can be regrouped while preserving the order of their elements. An identity element and inverse elements are not required.

For example, the positive integers with addition form a semigroup, since $(a+b)+c=a+(b+c)$ and a sum of positive integers is positive. They have no identity element: it would have to be zero, which is outside that set. Nonempty words over an alphabet with concatenation give a noncommutative example without an identity element. Including the empty word instead gives a free monoid.

The usefulness of a semigroup is that an ordered list has one product regardless of its parentheses. This permits replacing a consecutive block by its already computed value. Here the empty set is also allowed as a semigroup; a monoid is necessarily nonempty because it contains an identity element.

For an additive semigroup $D$ with identity zero, the semigroup algebra $k[D]$ has a $k$-basis of formal symbols $t^q$, $q\in D$, and multiplication $t^q{t}^{q^{\prime }}=t^{q+q^{\prime }}$. For $D=\mathbb{Z}[1/2{]}_{\ge 0}$ it is the integral domain ${\bigcup }_{h\ge 0}k[t^{1/2^h}]$. Its positive-exponent ideal is a nonzero idempotent ideal, and the strictly ascending chain $(t)⊊(t^{1/2})⊊\cdots$ shows that the algebra is not a Noetherian ring.

A semigroup presentation specifies generators $X$ and equalities between nonempty words. The resulting semigroup is the quotient of the free semigroup $X^{+}$ by the smallest congruence containing these equalities. Two words are equal precisely when one can be transformed into the other by finitely many replacements in contexts, using the defining equalities in either direction.

For a finite commuting family in a unital Banach algebra, divide each $x_j$ by $r(x_j)+\epsilon /2$. The spectral radius formula makes each normalized element's powers bounded. Commutativity lets every product be reordered into one power of each generator, so the entire generated semigroup is bounded. Renorming a bounded multiplicative semigroup gives the displayed bounds simultaneously. Applying this to two commuting elements proves submultiplicativity and subadditivity of their spectral radius.

A subsemigroup is a nonempty subset of a semigroup closed under its associative operation. In a compact Hausdorff space carrying a left-topological semigroup, descending chains of nonempty compact subsemigroups have a nonempty compact subsemigroup as their intersection. This supplies the minimal object used in the Ellis–Numakura lemma.

A topological semigroup is a semigroup with a topology for which multiplication is jointly continuous.