Let $D^2$ denote the unit disk in the plane, and let $C$ denote the set of (continuous) self-maps of $D^2$. Using $\circ$, as is common, to denote the binary operation on $C$ that takes a pair of continuous functions to another continuous function, we study some properties of the following algebras and their subalgebras:
\(\mathbb{A}=\langle D^2; C\rangle,\) and \(\mathbb{F}=\langle C; \circ\rangle.\) The algebra $\mathbb{A}$ can be naturally endowed with a topology, and we will suppress any notation the choice of topology on it, because we are interested in only the usual topology, so we think of $\mathbb{A}$ as a topological algebra; it is a {\bf multi-unary topological algebra} on the continuum $D^2$. As is well-known, there are various reasonable topologies that can be given to the algebra $\mathbb{F}$, but we will treat it only as an algebra for now. Note that the algebra $\mathbb{F}$ is a semigroup, and it is a subalgebra of a function algebra (an algebra of functions over a set that is closed under composition and contains the projection functions) over $D^2$.

Recall that in a semigroup, a left translation, $\lambda_a$ is a self-map of the semigroup defined using a parameter $a$, an element of the given semigroup, using the formula $\lambda_a(f)=af$. In our case, the parameters are continuous functions on $D^2$ and the semigroup operation is composition, so instead of juxtaposition of symbols, we will write $\lambda_a(f)=a\circ f$. Similarly, a right translation is defined by the other order of the ``multiplication’’: $\rho_a(f)=f\circ a$. Given an element $a\in C,$ we call the set \(\Lambda_a=\left\{(f,g)\in C^2\vert \lambda_a(f)=\lambda_a(g)\right\}=\left\{(f,g)\in C^2\vert a\circ f=a\circ g\right\}\) the {\bf kernel} of $\lambda_a$. kernels of right translations are defined similarly:
\({\rm P}_a=\left\{(f,g)\in C^2\vert \rho_a(f)=\rho_a(g)\right\}=\left\{(f,g)\in C^2\vert f\circ a=g\circ a\right\}.\) These are {\bf congruences} of the algebra (semigroup) $\mathbb{F}$; i.e. they are equivalence relations $\theta$ on the set $C$ that are compatible with the semigroup operation (composition). The compatibility property can be described via the containment ${(b\circ f,c\circ g)\vert (b,c),(f,g)\in\theta}\subseteq\theta$. On a set $X$, two special equivalence relations are congruences for any structure on $X$, namely the {\bf identity relation}, $\Delta_X={(p,p)\vert p\in X}$, and the {\bf all relation}, $\nabla_X={(p,q)\vert p, q\in X}$. It is clear that all kernels of left translations on a semigroup are congruences on that semigroup, and similarly, kernels of rigjt translations on a semigroup are congruences on that semigroup.

We will sketch a proof of the following:

Theorem. The algebra $\mathbb{F}$ has only three kinds of congruences, namely the identity relation, the all relation, and kernels of left translations by members of $C$.

Our proof of the above result will employ nonstandard methods and results from the theory of function algebras on finite sets, and interestingly, the above immediately entails the below consequence

Corollary. In the semigroup $\mathbb{F}$, every right translation equalizer is a left translation equalizer, and vice versa.

This is joint work with Malgorzata Marciniak (mmarciniak@lagcc.cuny.edu)

Some useful background material includes the following: 1. Dietlinde Lau, Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory. DOI: https://doi.org/10.1007/3-540-36023-9, Springer-Verlag Berlin Heidelberg 2006. ISBN: 978-3-540-36022-3, pgs 233-285. 2. Peter A. Loeb and Manfred P. H. Wolff, Nonstandard Analysis for the Working Mathematician. https://doi.org/10.1007/978-94-017-7327-0, Springer Science+Business Media Dordrecht 2015. ISBN: 978-94-017-7326-3, pgs. 165-176