Answer to Question #270886 in Abstract Algebra for kd bhai

Question #270886

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation

defined by aOb = max{ a,b}.


1
Expert's answer
2021-12-19T13:15:53-0500

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\\}\n\\\\=\\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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