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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p0735902.png" />-polygon, operand''
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''$R$-polygon, operand''
 
 
A non-empty set with a [[Monoid|monoid]] of operators. More precisely, a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p0735903.png" /> is called a left polygon over a monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p0735904.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p0735905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p0735906.png" /> the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p0735907.png" /> is defined, such that
 
 
 
<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/p/p073/p073590/p0735908.png" /></td> </tr></table>
 
  
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A non-empty set with a [[monoid]] of operators. More precisely, a non-empty set $A$ is called a left polygon over a monoid $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that
 +
$$
 +
\lambda (\mu a) = (\lambda\mu) a
 +
$$
 
and
 
and
 
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$$
<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/p/p073/p073590/p0735909.png" /></td> </tr></table>
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1a = a
 
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$$
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359012.png" />. A right polygon is defined similarly. Specifying an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359013.png" />-polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359014.png" /> is equivalent to specifying a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359015.png" /> from the monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359016.png" /> into the monoid of mappings of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359017.png" /> into itself that transforms 1 to the identity mapping. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359018.png" /> if and only if
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for any $\lambda, \mu \in R$, $a \in A$. A right polygon is defined similarly. Specifying an $R$-polygon $A$ is equivalent to specifying a homomorphism $\phi$ from the monoid $R$ into the monoid of mappings of the set $A$ into itself that transforms 1 to the identity mapping. Here $\lambda a = b$ if and only if
 
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$$
<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/p/p073/p073590/p07359019.png" /></td> </tr></table>
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\phi(\lambda)a = b$ \ .
 
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$$
 
In particular, each non-empty set may be considered as a polygon over the monoid of its mappings into itself. Therefore, polygons are closely related to the representation of semi-groups by transformations: cf. [[Transformation semi-group]].
 
In particular, each non-empty set may be considered as a polygon over the monoid of its mappings into itself. Therefore, polygons are closely related to the representation of semi-groups by transformations: cf. [[Transformation semi-group]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359020.png" /> is a [[Universal algebra|universal algebra]] whose signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359021.png" /> contains only unary operations, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359022.png" /> can be converted into a polygon over the free monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359023.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359024.png" /> by putting
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If $A$ is a [[universal algebra]] whose signature $\Omega$ contains only unary operations, then $A$ can be converted into a polygon over the free monoid $F$ generated by $\Omega$ by putting
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359025.png" /></td> </tr></table>
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f_1 \cdots f_n a = f_1(\cdots(f_n(a)\cdots)
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359028.png" /> is the set of input signals for an automaton having set of states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359030.png" /> is similarly transformed into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359031.png" />-polygon (cf. [[Automata, algebraic theory of|Automata, algebraic theory of]]).
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for any $f_i \in \Omega$, $a \in A$. If $\Omega$ is the set of input signals for an automaton having set of states $A$, then $A$ is similarly transformed into an $F$-polygon (cf. [[Automata, algebraic theory of]]).
  
 
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359032.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359033.png" />-polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359034.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359035.png" />-polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359036.png" /> is called a homomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359037.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359039.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359040.png" /> one arrives at the definition of an endomorphism. All endomorphisms of a polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359041.png" /> form a monoid, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359042.png" /> can be considered as a polygon over it.
 
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359032.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359033.png" />-polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359034.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359035.png" />-polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359036.png" /> is called a homomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359037.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359039.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359040.png" /> one arrives at the definition of an endomorphism. All endomorphisms of a polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359041.png" /> form a monoid, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073590/p07359042.png" /> can be considered as a polygon over it.

Revision as of 21:19, 19 December 2015

$R$-polygon, operand

A non-empty set with a monoid of operators. More precisely, a non-empty set $A$ is called a left polygon over a monoid $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that $$ \lambda (\mu a) = (\lambda\mu) a $$ and $$ 1a = a $$ for any $\lambda, \mu \in R$, $a \in A$. A right polygon is defined similarly. Specifying an $R$-polygon $A$ is equivalent to specifying a homomorphism $\phi$ from the monoid $R$ into the monoid of mappings of the set $A$ into itself that transforms 1 to the identity mapping. Here $\lambda a = b$ if and only if $$ \phi(\lambda)a = b$ \ . $$ In particular, each non-empty set may be considered as a polygon over the monoid of its mappings into itself. Therefore, polygons are closely related to the representation of semi-groups by transformations: cf. Transformation semi-group.

If $A$ is a universal algebra whose signature $\Omega$ contains only unary operations, then $A$ can be converted into a polygon over the free monoid $F$ generated by $\Omega$ by putting $$ f_1 \cdots f_n a = f_1(\cdots(f_n(a)\cdots) $$

for any $f_i \in \Omega$, $a \in A$. If $\Omega$ is the set of input signals for an automaton having set of states $A$, then $A$ is similarly transformed into an $F$-polygon (cf. Automata, algebraic theory of).

A mapping of an -polygon into an -polygon is called a homomorphism if for any and . For one arrives at the definition of an endomorphism. All endomorphisms of a polygon form a monoid, and can be considered as a polygon over it.

An equivalence on an -polygon is called a congruence if implies for any . The set of congruence classes of is naturally transformed into an -polygon, called a quotient polygon of the polygon and denoted by . If is a polygon over , then in one can define a relation by putting if for all . The relation is a congruence on the monoid , and is transformed in a natural fashion into a polygon over the quotient monoid . If the polygon arose from a certain automaton, then this transition is equivalent to identifying identically acting sequences of input signals. In universal algebra one considers the usual constructions of direct and subdirect product, but in addition in polygon theory one may consider a wreath product construction important in the algebraic theory of automata. The free product (or co-product) of polygons coincides with their disjoint union.

A polygon may be regarded as a non-additive analogue of a module over a ring, which serves as a rich source of problems in the theory of polygons. In particular, a relationship has been established between polygons and radicals in semi-groups (cf. Radical in a class of semi-groups), and studies have been made on the relation between the properties of a monoid and those of polygons over them. For example, all left -polygons are projective if and only if is a one-element group, while the injectivity of all polygons over a commutative monoid is equivalent to the presence in of a zero and the generation of all its ideals by idempotents (see Homological classification of rings).

If is a monoid with zero 0, one can speak of an -polygon with a zero as an -polygon with a distinguished point satisfying for all . The theory of polygons containing zeros has some special features.

Every polygon can be considered as a functor from a one-object category into the category of sets.

References

[1] M.A. Arbib (ed.) , Algebraic theory of machines, languages and semigroups , Acad. Press (1968)
[2] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967)
[3] L.A. Skornyakov, "Generalizations of modules" , Modules , 3 , Novosibirsk (1973) pp. 22–27 (In Russian)
[4] L.A. Skornyakov, A.V. Mikhalev, "Modules" Itogi Nauk. i Tekhn. Alg. Topol. Geom. , 14 (1976) pp. 57–190 (In Russian)


Comments

In the West, left polygons over a monoid $M$ are usually called $M$-sets; the term "operand" is also in use. The category of all $M$-sets ($M$ fixed) forms a topos; but for this it is essential not to exclude (as above) the empty $M$-set.

Without assuming commutativity as above, the monoids all of whose non-empty left polygons (or, all of whose pointed left polygons) are injective have a few characterizations, which are reviewed in [a3]. As noted above, there are no non-trivial monoids all of whose left polygons are projective, but the perfect monoids, defined (like perfect rings, cf. Perfect ring) by every left polygon having a projective covering, are non-trivial; see [a1], [a2].

References

[a1] J. Fountain, "Perfect semigroups" Proc. Edinburgh Math. Soc. , 20 (1976) pp. 87–93
[a2] J. Isbell, "Perfect monoids" Semigroup Forum , 2 (1971) pp. 95–118
[a3] R.W. Yoh, "Congruence relations on left canonic semigroups" Semigroup Forum (1977) pp. 175–183
[a4] S. Eilenberg, "Automata, languages and machines" , Acad. Press (1974)

Comments

The terms monoid action or monoid act, or action of a monoid on a set are also common, as is act for the set acted on; see [b1].

References

[b1] Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev, Monoids, Acts and Categories: With Applications to Wreath Products and Graphs, Walter de Gruyter (2000) ISBN 3110812908
How to Cite This Entry:
Polygon (over a monoid). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polygon_(over_a_monoid)&oldid=37002
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article