# Monogenic semi-group

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A semi-group generated by one element. The monogenic semi-group generated by an element is usually denoted by (sometimes by ) and consists of all powers with natural exponents. If all these powers are distinct, then is isomorphic to the additive semi-group of natural numbers. Otherwise is finite, and then the number of elements in it is called the order of the semi-group , and also the order of the element . If is infinite, then is said to have infinite order. For a finite monogenic semi-group there is a smallest number with the property , for some ; is called the index of the element (and also the index of the semi-group ). In this connection, if is the smallest number with the property , then is called the period of (of ). The pair is called the type of (of ). For any natural numbers and there is a monogenic semi-group of type ; two finite monogenic semi-groups are isomorphic if and only if their types coincide. If is the type of a monogenic semi-group , then are distinct elements and, consequently, the order of is ; the set is the largest subgroup and smallest ideal in ; the identity of the group is the unique idempotent in , where for any such that ; is a cyclic group, a generator being, for example, . An idempotent of a monogenic semi-group is a unit (zero) in it if and only if its index (respectively, period) is equal to 1; this is equivalent to the given monogenic semi-group being a group (respectively, a nilpotent semi-group). Every sub-semi-group of the infinite monogenic semi-group is finitely generated.