Submonoid

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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ANALOGY: This is an analogue in monoid of a property encountered in group. Specifically, it is a submonoid property analogous to the subgroup property: subgroup
View other analogues of subgroup | View other analogues in monoids of subgroup properties (OR, View as a tabulated list)

Definition

Definition in terms of closure under binary operation

This definition of submonoid corresponds to the textbook definition of monoid.

Let $M$ be a monoid. A subset $N$ of $M$ is termed a submonoid if the following two conditions hold:

• Whenever $a,b$ belong to $N$, the product $ab$ belongs to $N$.
• With this induced multiplication, $N$ becomes a group in its own right (i.e., it has an identity element). Note that associativity in $N$ follows automatically from associativity in $M$.

Normal submonoids

A submonoid $H$ of a monoid $M$ is said to be a normal submonoid if $Hx=xH\forall x\in M$.