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Archangel Macsika

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation defined by aOb = max{ a,b}.

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A semigroup is a pair (S,\circ), where S is a non-empty set and \circ:S\times S\to S is an associative binary operation on S. A monoid is a semigroup (S,\circ) with identity element e\in S in the sence that e\circ s=s\circ e=s for any s\in S.


Let us show that the set of positive integer \N is a monoid for the operation defined by a\circ b = \max\{ a,b\}.

If a,b\in \N then a\circ b = \max\{ a,b\}\in\N, and hence the operation is defined on the set \N.

Since


a\circ(b\circ c)=a\circ\max\{b, c\}=\max\{a,\max\{b, c\}\}=\max\{a,b, c\} \\=\max\{\max\{a,b\}, c\}=\max\{a\circ b, c\}=(a\circ b)\circ c


for any a,b,c\in\N, we conclude that operation \circ is associative, and hence (\N,\circ) is a semigroup.


Taking into account that a\circ 1=\max\{a,1\}=a=\max\{1,a\}=1\circ a for each a\in\N, we conclude that 1 is the identity of the semigroup (\N,\circ), and consequently (\N,\circ) is a monoid.

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