View source for Semi-group algebra

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An
[[Algebra|algebra]] $\Phi(S)$ over a field $\Phi$ with a basis $S$ that is at the same time a multiplicative
[[Semi-group|semi-group]]. In particular, if $S$ is a group, one obtains a
[[Group algebra|group algebra]]. If the semi-group $S$ contains a [[zero]], this zero is usually identified with the zero of the algebra $\Phi(S)$. The problem of describing all linear representations of a semi-group $S$ over a field $\Phi$ (cf. [[Linear representation]]; [[Representation of a semi-group]]) is equivalent to that of describing all representations of the algebra $\Phi(S)$. The importance of semi-group algebras in the theory of semi-groups is the possibility they offer of utilizing the richer tools of the theory of algebras to study linear representations of semi-groups. An example of this kind of result is: The algebra $\Phi(S)$ of a finite semi-group $S$ is semi-simple if and only if all linear representations of the semi-group $S$ over the algebra $\Phi$ are reducible.

====Comments====
More precisely, let $S$ be a semi-group and $\Phi$ a field. Consider the vector space $V$ of all formal finite sums $V=\{\sum_{s\in S} a_s s\}$, i.e. the vector space over $\Phi$ with basis $S$. The semi-group multiplication $(s,t)\mapsto st$ extends linearly to define an algebra structure on $V$. This is the semi-group algebra $\Phi[S]$.

If $S$ is a semi-group with zero $z$, the subspace $\Phi_z$ is an ideal in $\Phi[S]$ and the contracted semi-group algebra $\Phi_0[S]$ is the quotient algebra $\Phi_0[S] = \Phi[S]/\Phi_z$.

For an inverse semi-group (cf. [[Inversion semi-group]]) one has the following analogue of Maschke's theorem (cf. [[Group algebra]]). The semi-group algebra $\Phi[S]$ of a finite inverse semi-group $S$ is semi-simple if and only if the characteristic of $\Phi$ is zero or is a prime that does not divided the order of any sub-semi-group of $S$.

====References====
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|valign="top"|{{Ref|ClPr}}||valign="top"|  A.H. Clifford,   G.B. Preston,   "The Algebraic theory of semigroups", '''1''', Amer. Math. Soc.  (1961)        {{MR|0132791}}  {{ZBL|0111.03403}} 
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