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)

Comments

Regular varieties of unary algebras can be characterized in categorical terms, as those for which the forgetful functor to the category of sets preserves coproducts (that is, such that the coproduct of a family of algebras in the variety is carried by the disjoint union of their carrier sets). The semi-group associated with such a variety can also be recovered categorically, without having recourse (as in the main article above) to a specific presentation of the variety by operations and identities: it is the semi-group of endomorphisms of the forgetful functor from the variety to sets.