# Unary algebra

unoid

A universal algebra with a family of unary operations . An important example of a unary algebra arises from a group homomorphism from an arbitrary group into the group of all permutations of a set . Such a homomorphism is called an action of the group on . The definition, for each element , of a unary operation as the permutation in corresponding to the element under the homomorphism yields a unary algebra , in which

Every module over a ring carries a unary algebra structure. Every deterministic semi-automaton (cf. Automaton, algebraic theory of) with set of states and input symbols may also be considered as a unary algebra , where is the state onto which the state is mapped by the action of the input symbol .

A unary algebra with a single basic operation is called mono-unary, or a unar. An example of a unar is the Peano algebra , where and .

The identities of an arbitrary unary algebra can only be of the following types:

. ,

. ,

. ,

. ,

. ,

. .

The identity is equivalent to , being satisfied only by a -element algebra. A variety of unary algebras defined only by identities of the form , or is said to be regular. There exists the following link between regular varieties of unary algebras and semi-groups (cf. [1], [3], [4]).

Let be a regular variety of unary algebras given by a set , , of function symbols and a set of identities. Each symbol corresponds to an element , and for every identity of the form from one writes the defining relation

Let be the semi-group with generators , , and the above defining relations, and let be the semi-group with an identity adjoined. For every relation of the form in (if they are any) one writes the defining relation as . The semi-group obtained from by adjoining these defining relations is said to be associated with the variety . There are many ways of characterizing this variety. If contains only identities of the form , then one may restrict oneself to the construction of . By defining a unary operation in one obtains a unary algebra , which is a -free algebra of rank 1. The group of all automorphisms of the unary algebra is isomorphic to the group of invertible elements of the semi-group .

#### References

 [1] A.I. Mal'tsev, "Algebraic systems" , Springer (1972–1973) (Translated from Russian) [2] G. Birkhoff, T. Bartee, "Modern applied algebra" , McGraw-Hill (1970) [3] D.M. Smirnov, "Regular varieties of algebras" Algebra and Logic , 15 : 3 (1976) pp. 207–213 Algebra i Logika , 15 : 3 (1976) pp. 331–342 [4] D.M. Smirnov, "Correspondence between regular varieties of unary algebras and semigroups" Algebra and Logic , 17 : 4 (1978) pp. 310–315 Algebra i Logika , 17 : 4 (1978) pp. 468–477 [5] B. Jónsson, "Topics in universal algebra" , Springer (1972)