I call ** adjunction algebra** a universal algebra with one binary operation denoted as the punctuation sign (

**;**) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going against the regular punctuation practice and creating typographical problems), and 3 axioms which are universal closures of these formulas (between brackets I indicate how I refer to them):

$((x; y); y) = (x; y)$ [right idempotency]

$((x; y); z) = ((x; z); y)$ [right commutativity]

$((x; y); z) = (x; y) \rightarrow ((x; z) = x \vee z = y)$ [right atomicity]

I call *normal* or *normalized* adjunction algebra, an algebra which additionally to adjunction has a constant $0$ such that:

- $(0; x) \ne x$

I think, this algebra is important for algebraization of set theory. Also, for several years now, I am using it, in another presentation, in my research of semantics of natural languages. I used to call the operation (;) "qualification", because it explains the semantics of qualified names in computer science. Lately, I learnt that this operation is called "adjunction" from @Joel's answer to my question. The axioms above are from the paper by Kirby referenced there. True, Kirby does not call this "adjunction algebra", because along these axioms Kirby also uses an induction axiom and he studies a "theory", not an "algebra".

The intended interpretation in set theory of the $(x; y)$ is $x \cup ${$y$}, and in this interpretation $(x; y) = x$ iff $y \ \epsilon \ x$. This shows why adjunction is important for set theory - set theory can be presented in the language of one operation. In Kirby's paper the operation of union is defined by induction (which is an axiom) and it sounds to me improbable that it can be defined through ";" an "0" without such an axiom.

Here are some of my questions:

(1) Does the axioms above occur in abstract algebra and what are the names of the properties they express?

(2) Is what I call normal adjunction algebra "free" in some sense (my algebraic background is not very strong, so my question might not be precise).

(3) Are there algebras where the induction principle, in any of its different forms, is represented algebraically (i.e. as identities or quasiidentities)?