Unary algebra

Definition

Let $A$ be a set and let $I$ be an index set. A unary algebra is a pair $(A, (f_i)_{i \in I})$ , where each $f_i \colon A \to A$ is a unary operation on $A$ . It is mono-unary when $I$ has exactly one element. A homomorphism of unary algebras $(A, (f_i))$ and $(B, (g_i))$ with the same index set is a function $h \colon A \to B$ with $h(f_i(a)) = g_i(h(a))$ for all $i \in I$ and $a \in A$ .

Remarks

Example: $(\Z, s)$ with $s(n) = n + 1$ is a mono-unary algebra, since $s$ maps $\Z$ to $\Z$ . The map $h \colon \Z \to \Z/2\Z$ , $h(n) = n \bmod 2$ , is a homomorphism to $(\Z/2\Z, t)$ with $t(x) = x + 1$ , because $h(n + 1) = h(n) + 1$ .

Non-example: $(\N, p)$ with the predecessor $p(n) = n - 1$ is not a unary algebra, since $p(0) = -1 \notin \N$ ; the predecessor is only a partial operation on $\N$ .

A unary algebra $(A, (f_i)_{i \in I})$ is the same as an action of the free monoid $I^*$ on $A$ : a word $i_1 i_2 \cdots i_k$ acts by $f_{i_k} \circ \cdots \circ f_{i_1}$ in one common convention.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Element definition
Function concept
Unary operation concept
Set definition
Unary algebra definition

Tag	`MS5PXY4D9Y`
URI	`m2://MS5PXY4D9Y`
Kind	definition
Writing	W0 stub
MSC	08A60
Defines	Unary algebra
Uses	Set, Unary operation, Function

Unary algebra

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here