# What kind of algebraic structure is the one with NAND?

In her book Model Theory, María Manzano mentions the following kinds algebraic structures:

1. Group
2. Rings and fields
3. Order
4. Well-order
5. Peano structures
6. Boolean algebras
7. Bolean rings

Nevertheless, it's not clear to me what kind of structure would $$\mathfrak{W} = ⟨W, \uparrow⟩$$ be, where:

• $$W = \{0, 1\}$$

And the following axiom is fulfilled whenever $$p$$ and $$q$$ range over $$W$$:

• $$((p \uparrow q) \uparrow r) \uparrow (p \uparrow ((p \uparrow r) \uparrow p)) = r$$ (Aka: Wolfram's axiom)

It's relatively easy to prove that there is an isomorphism between $$\mathfrak{W}$$ and a Boolean structure when we consider the following equivalences:

• $$p \uparrow p \iff -p$$
• $$p \sqcup q \iff (p \uparrow q) \uparrow (p \uparrow q)$$
• $$p \sqcap q \iff (p \uparrow p) \uparrow (q \uparrow q)$$

I suppose, though, that doesn't automatically make $$\mathfrak{W}$$ a Boolean structure. It's possible that there is no name for this kind of structure, but that would be kind of awful since we can define every possible function of classical propositional logic from $$\uparrow$$.

• I think the closest it has to a name is that it's a magma satisfying Wolfram's axiom. It is standard for there to be many mutually interdefinable ways of characterising essentially the same class of algebraic structures. E.g., groups can be characterised over the signature $(*)$ where $x *y$ denotes what we would usually write as $xy^{-1}$. Commented Feb 6, 2018 at 20:25