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Let there be an algebra $(S,f,t)$ with the laws:

$$ f(t(x)) = t(x) \\ t(f(x)) = t(x) $$

or, put another way,

$$ f \circ t = t \\ t \circ f = t. $$

  1. Does that particular algebra have a name?
  2. Does a set with two unary operations (algebras whose signature is (1,1) in general) have a name?

The closest thing I found (by replacing $f$ with $x$ and $\circ$ with $\vee$) was null semigroup such as $(S,\vee,t)$ where

$$ x \vee t = t \\ t \vee x = t $$

(but without the idempotency law $x \vee x = x$, in case $\vee$ reminds you of lattices). But the signature of such null semigroup is (2,0), not (1,1).

edom
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  • Not an answer, but a class of natural models: suppose $\tau_1$ and $\tau_2$ are topologies on a set $X$ with $\tau_1$ coarser than $\tau_2$ (that is, all $\tau_1$-open sets are $\tau_2$-open). Then letting $A$ be the algebra with underlying set $\mathcal{P}(X)$, $f$ interpreted as $\tau_2$-closure, and $t$ interpreted as $\tau_1$-closure, yields a model. Of course, this doesn't actually characterize your theory since e.g. in these models $f$ and $t$ will always be idempotent, but it's a natural class. – Noah Schweber Jun 02 '16 at 20:59
  • A set with two unary operations is a bi-unary algebra (see, for example, Burris & Sankappanavar, II.8, exercise 4). In general, an algebra, all of its operations are unary is just a unary algebra, and the most notable class is the one of mono-unary algebras (only one such opration). This answers your second question; about the first, I don't know anything. – amrsa Jun 26 '16 at 11:46

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