I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. The philosophy behind this is to see algebra as the study of relations, that is, of at least binary laws. We therefore impose that there is no $0-ary$ or $1-ary$ relations in its signature (if we want to use this concept to formalize what's a structure). For simplicity, I'll only focus on binary laws.

Let $A$ be some set and let $.$ be some binary law $.$ : $A \times A \rightarrow A$. We say that an element $1_A$ of $A$ is a unit relative to $A$ if for all $x$, in $A$, $1_A.x = x.1_A = x$. A structure $(A,.)$ that admits a unit relative to $A$ is called unital.

Let $B$ be a subset of $A$. We say that $(B, ._B)$ is a supstructure of $A$ if it is closed under $._B$, the restriction of $.$ to $B$. Such supstruture is unital if there exists some element $1_B$ in $B$ such that for all $x$ in $B$, $1_B . x = x. 1_B = x$.

We now use the word magna, monoid, ring, $\ldots$ with their common use. Example: Let the ordinal $\omega n$, and define a binary law $+$ such that if $x$ is in $\omega m$ but not in $\omega p$ with $m > p$, then for any $y$ in $\omega p$, $x+ y = x$. That is, $x$ absorbs the elements that are of lower infinite degree. The addition is defined as on $\mathbb{N}$ for elements of the same infinite degree. Clearly, $\omega n$ has $n$ relative units. This clearly gives non trivial example of supmonoids of some monoid (there are a lots of them).

A morphism of "struture" is any $f : (A,.) \rightarrow (B,.)$ such that for all $x, y$, $f(x.y) = f(x).f(y)$. We don't require that the global units are preserved. A morphism $f : (A,.) \rightarrow (B,.)$ can therefore map $A$ to a non trivial supstructure of $B$.

A structure $A$ is said to be self-similar if it is isomorphic to one of its non trivial supstructure $B$. Clearly, if this happens, there exists an infinity of such isomorphisms. Example: $(\mathbb{N},+)$ seen as a monoid is obviously isomorphic to the even numbers or any $k \mathbb{N}$. The field of complex numbers admits such isomorphism too.

Let $B$ be a non trivial supstructure in $A$. Such supstructure is said to repeat itself in $A$ if there exists an isomorphism from $B$ to another supstruture of $A$ disjoint from $B$. If $B$ is moreover self-similar, we say that $B$ is repeatedly self-similar. Example: For $(\omega n,+)$, $\omega$ is repeatedly self-similar. (We could also define its indice in $A$, which would be $n$)

If $+$ is commutative, we say that $x$ $+$-absorbs $y$ if $x + y = x$. We define the antonymic law $\#$ associated to $+$ as follow :

$x$ $\#$-absorbs $y$ iff $y$ $+$-absorbs $x$ (in other words, $x \# y = x$ iff $x + y = y$

$x \# y = x + y$ otherwise.

Clearly, if $+$ is commutative, $\#$ is commutative. If $1$ is a unit relative to $(A,+)$, then clearly $1$ is an absorbing element relative to $(A,\#)$. If $+$ is associative, then $\#$ is associative too (case by case check). This shows a new kind of duality neutral element $\leftrightarrow$ absorbing elements. Indeed a neutral element is an element absorbed by everyone, hence the name antonymic law. Let me add that algebraically speaking, the only ``ontology'' one can give to neutral elements or absorbing elements is that they are idempotent. Any idempotent element can be seen as a relative neutral or relative absorbing element of some (not necessarily non trivial) supstructure.

We define an infmonoid to be the dual of a monoid, that is a structure $(M,+,i)$ where

$+$ is associative,

$i$ is an absorbing element.

There are many such example of infmonoids. We could obviously always define the dual of any known algebraic stuctures, say, infgroups, $\ldots$ As a side remark, note that the concept of annulus $(M,+,.,0,i)$ defined as

$(M,+,0)$ is a commutative monoid,

$(M,.,i)$ is an infmonoid,

$.$ is distributive over $+$,

is in my opinion a much more elegant concept than the one of semi-ring (In particular, $i = 0$ iff $i$ is $+$-invertible). A field is a special case of annulus, and can (if I did no mistake) be formulated entirely algebraically (that is, it can be easily categorified). Some example of annulus are given by what I call the ``capped annulus'' $(C_{n+1}, \oplus, \otimes, n, 0)$ (finite counterpart of max-plus algebra) defined on ${0,1, \ldots, n}$ with $x \oplus y = min(x,y)$ and $x \otimes y = min(x \times y, n)$. One can easily define their additive counterpart.

Now, say that a structure $(A,.)$ is antonymic symmetric iff it is isomorphic to $(A,\#)$. Example: $(\mathbb{N} \cup \{\infty\},+)$ is clearly antonymic symmetric. The $2$-elements Boolean algebra seen as a lattice is antonymic symmetric ($\#$ is nothing else than the swap of operations). In particular, if $L$ is a totally ordered bounded lattice, then it is antonymic symmetric. An antonymic symmetric structure $(A,.)$ that is self-similar to one of its supstruture $(B,.)$ is called entirely self-similar. Example: $\omega^2 +1 $ is clearly entirely self-similar. Does anyone have any other example of entirely self-similar strutures ? In particular, non commutative ones ?