# Unary algebra

unoid

A universal algebra $\langle A , \{ {f _ {i} } : {i \in I } \} \rangle$ with a family $\{ {f _ {i} } : {i \in I } \}$ of unary operations $f _ {i} : A \rightarrow A$. An important example of a unary algebra arises from a group homomorphism $\phi : G \rightarrow S _ {A}$ from an arbitrary group $G$ into the group $S _ {A}$ of all permutations of a set $A$. Such a homomorphism is called an action of the group $G$ on $A$. The definition, for each element $g \in G$, of a unary operation $f _ {g} : A \rightarrow A$ as the permutation $\phi ( g)$ in $S _ {A}$ corresponding to the element $g$ under the homomorphism $\phi$ yields a unary algebra $\langle A , \{ {f _ {g} } : {g \in G } \} \rangle$, in which

$$f _ {1} ( x) = x ,\ \ f _ {g} ( f _ {h} ( x) ) = f _ {gh} ( x) ,\ x \in A ,\ \ g , h \in G .$$

Every module over a ring carries a unary algebra structure. Every deterministic semi-automaton (cf. Automaton, algebraic theory of) with set $S$ of states and input symbols $a _ {1} \dots a _ {n}$ may also be considered as a unary algebra $\langle S , f _ {1} \dots f _ {n} \rangle$, where $f _ {i} ( s) = a _ {i} s$ is the state onto which the state $s$ is mapped by the action of the input symbol $a _ {i}$.

A unary algebra with a single basic operation is called mono-unary, or a unar. An example of a unar is the Peano algebra $\langle P , f \rangle$, where $P = \{ 1 , 2 ,\dots \}$ and $f ( n) = n + 1$.

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

$\textrm{ I } _ {1}$. $f _ {i _ {1} } \dots f _ {i _ {k} } ( x) = f _ {j _ {1} } \dots f _ {j _ {l} } ( x)$,

$\textrm{ II } _ {1}$. $f _ {i _ {1} } \dots f _ {i _ {k} } ( x) = f _ {j _ {1} } \dots f _ {j _ {l} } ( y)$,

$\textrm{ I } _ {2}$. $f _ {i _ {1} } \dots f _ {i _ {k} } ( x) = x$,

$\textrm{ II } _ {2}$. $f _ {i _ {1} } \dots f _ {i _ {k} } ( x) = y$,

$\textrm{ I } _ {3}$. $x = x$,

$\textrm{ II } _ {3}$. $x = y$.

The identity $\textrm{ II } _ {2}$ is equivalent to $\textrm{ II } _ {3}$, being satisfied only by a $1$- element algebra. A variety of unary algebras defined only by identities of the form $\textrm{ I } _ {1}$, $\textrm{ I } _ {2}$ or $\textrm{ I } _ {3}$ is said to be regular. There exists the following link between regular varieties of unary algebras and semi-groups (cf. [1], [3], [4]).

Let $V$ be a regular variety of unary algebras given by a set $\{ {f _ {i} } : {i \in I } \}$, $I \neq \emptyset$, of function symbols and a set $\Sigma$ of identities. Each symbol $f _ {i}$ corresponds to an element $a _ {i}$, and for every identity of the form $\textrm{ I } _ {1}$ from $\Sigma$ one writes the defining relation

$$a _ {i _ {1} } \dots a _ {i _ {k} } = \ a _ {j _ {1} } \dots a _ {j _ {l} } .$$

Let $P$ be the semi-group with generators $a _ {i}$, $i \in I$, and the above defining relations, and let $P ^ {1}$ be the semi-group $P$ with an identity $e$ adjoined. For every relation of the form $\textrm{ I } _ {2}$ in $\Sigma$( if they are any) one writes the defining relation as $a _ {i _ {1} } \dots a _ {i _ {k} } = e$. The semi-group $P _ {V}$ obtained from $P ^ {1}$ by adjoining these defining relations is said to be associated with the variety $V$. There are many ways of characterizing this variety. If $\Sigma$ contains only identities of the form $\textrm{ I } _ {1}$, then one may restrict oneself to the construction of $P$. By defining a unary operation $f _ {i} ( x) = x a _ {i}$ in $P _ {V}$ one obtains a unary algebra $\langle P _ {V} , \{ {f _ {i} } : {i \in I } \} \rangle$, which is a $V$- free algebra of rank 1. The group of all automorphisms of the unary algebra $\langle P _ {V} , \{ {f _ {i} } : {i \in I } \} \rangle$ is isomorphic to the group $P _ {V} ^ {*}$ of invertible elements of the semi-group $P _ {V}$.

#### 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)