View source for Radical in a class of semi-groups

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771001.png" /> associating to each [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771002.png" /> a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771003.png" /> (cf. [[Congruence (in algebra)|Congruence (in algebra)]]) and having the following properties: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771004.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771006.png" /> (0 denotes the equality relation), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771007.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771008.png" /> is a congruence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r0771009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710011.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710012.png" />. If 1) and 3) are satisfied, then 2) is equivalent to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710013.png" /></td> </tr></table>

for every congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710015.png" />. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710016.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710018.png" />-semi-simple if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710019.png" />. The class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710020.png" />-semi-simple semi-groups contains the one-element semi-group and is closed relative to isomorphism and subdirect products. Conversely, each class of semi-groups having these properties is the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710021.png" />-semi-simple semi-groups for some radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710024.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710026.png" />-radical. In contrast to rings, in semi-groups the radical is not determined by the corresponding radical class. If in the definition of a radical the discussion is limited to congruences defined by ideals, then another concept of a radical arises, where the corresponding function chooses an [[Ideal|ideal]] in each semi-group.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710027.png" /> is a class of semi-groups that is closed relative to isomorphisms and that contains the one-element semi-group, then the function that associates to each semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710028.png" /> the intersection of all congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710030.png" /> turns out to be a radical, called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710031.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710032.png" /> coincides with the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710033.png" />-semi-simple semi-groups if and only if it is closed relative to subdirect products. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710034.png" /> is the largest quotient semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710035.png" /> that lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710036.png" /> (see [[Replica|Replica]]).

Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710037.png" /> be the class of semi-groups admitting a faithful irreducible representation (cf. [[Representation of a semi-group|Representation of a semi-group]]). Then

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where

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Radicals defined on a given class of semi-groups that is closed relative to homomorphic images have been studied.

Related to each radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710041.png" /> is the class of left polygons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710042.png" /> (cf. [[Polygon (over a monoid)|Polygon (over a monoid)]]). Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710043.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710044.png" />-polygon, then a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710046.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710048.png" />-annihilating if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710049.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710050.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710051.png" />. The least upper bound of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710052.png" />-annihilating congruences turns out to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710053.png" />-annihilating congruence, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710054.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710055.png" />, by definition, consists of all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710056.png" />-polygons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710059.png" /> runs through the class of all semi-groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710060.png" /> is a congruence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710061.png" />, then a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710062.png" />-polygon lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710063.png" /> if and only if it lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710064.png" /> when considered as a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710065.png" />-polygon. Conversely, if one is given a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710066.png" /> of left polygons with these properties and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710067.png" /> is the class of all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710068.png" />-polygons in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710069.png" />, then the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710070.png" /></td> </tr></table>

is a radical.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "The algebraic theory of semi-groups" , '''2''' , Amer. Math. Soc.  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,   "Radicals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077100/r07710071.png" />-rings" , ''Selected problems in algebra and logic'' , Novosibirsk  (1973)  pp. 283–299  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.H. Clifford,   "Radicals in semigroups"  ''Semigroup Forum'' , '''1''' :  2  (1970)  pp. 103–127</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.N. Roiz,   B.M. Schein,   "Radicals of semigroups"  ''Semigroup Forum'' , '''16''' :  3  (1978)  pp. 299–344</TD></TR></table>