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A lattice is a set $L$ with two binary operations, $\lor$ "join" and $\land$ "meet". In a complemented lattice, for every element $a$ there exists an element $a^{\perp}$ such that $a \lor a^\perp=1$ and $a \land a^\perp=0$. Note this "complement" element need not be unique. But if it is unique, as is true in distributive lattices, the lattice is called a uniquely complemented lattice.

It seems to me that in a uniquely complemented lattice, each element having a unique "complement" element could define a unary operation $\perp: L \to L$ defined by $\perp(a)=a^\perp$. But I have never seen it developed this way (the way the additive inverse is defined as unary operation in a group or the multiplicative inverse is defined as a unary operation in a field). Is there a problem with defining a uniquely complemented lattice this way: that is, a lattice with a unary operation $\perp$ in addition to its two binary operations $\lor$ and $\land$?

Along the same lines, is it OK to define $0$ and $1$ as nullary operations on the lattice, the same way $0$ and $1$ are considered nullary operations on a ring? Although it seems correct to me, I could be missing something. Have nullary and unary operations been described this way in lattice theory before? Thanks for reading!

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Nullary (constants) and unary operations are perfectly fine in universal algebra. And indeed, top and bottom elements in a bounded lattice are defined using constants. It's a little bit surprising that you didn't meet that definition yet; actually the phrase “in the language of bounded lattices” implicitly says that you have both 0 and 1 in your signature.

I don't exactly remember the treatment there, but it is for sure that in B&S A Course in Universal Algebra this is explained this way.

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  • $\begingroup$ Thanks for your answer, Pedro! And thanks for the book you referenced - it looks like a great resource! $\endgroup$ – Mathemanic Sep 11 '15 at 19:45

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