nLab skew lattice

Idea

The notion of skew lattices is that of lattices (as an algebraic structure $(L, \wedge, \vee)$), except that commutativity axioms are dropped.

Definition

A skew lattice is a set $L$ equipped with two associative operations $\vee, \wedge$ that satisfy the following versions of absorption laws:

$$x \wedge (x \vee y) = x = (y \vee x) \wedge x, \qquad x \vee (x \wedge y) = x = (y \wedge x) \vee x \,.$$

Observe that the notion of skew lattice is self-dual: a statement in the language of skew lattices holds iff the dual statement, obtained by swapping all instances of $\wedge$ with $\vee$ and vice-versa, also holds.

Properties

Proposition

In a skew-lattice, every element $x$ satisfies $x \wedge x = x$. Dually, $x \vee x = x$.

Proof

We have

$$x = x \wedge (x \vee (x \wedge x)) = x \wedge x$$

where the first equation is from the first absorption law and the second equation is from the second absorption law.

It follows that a skew lattice is a band under either operation $\wedge$, $\vee$. Accordingly (see there), there is a natural preorder structure definable from either operation; call them $\preceq_\wedge$ and $\preceq_\vee$. These two preorder structures are opposite to one another, by the following results (as adapted from a relevant MathStackExchange discussion):

Lemma

In a skew lattice, $x = x \wedge y$ iff $y = x \vee y$. Similarly, $y = x \wedge y$ iff $x = x \vee y$.

Proof

Immediate from the absorption laws.

Proposition

In a skew lattice, we have $x \preceq_\wedge y$ iff $y \preceq_\vee x$.

Proof

The two relations mean, by definition, $x = x \wedge y \wedge x$ and $y = y \vee x \vee y$. We are to show that these are equivalent. By duality, only the forward implication needs to be proven.

By hypothesis $x = x \wedge (y \wedge x)$ and the lemma, we have $y \wedge x = x \vee (y \wedge x)$. Denote the last equation by $E$.

Then

$$\array{ y \vee x & = & (y \vee x) \wedge ((y \vee x) \vee (y \wedge x)) & (absorption) \\ & = & (y \vee x) \wedge (y \vee (x \vee (y \wedge x)) & (associativity\; of\; \vee) \\ & = & (y \vee x) \wedge (y \vee (y \wedge x)) & (by\; E) \\ & = & (y \vee x) \wedge y & (absorption) }$$

Then, from $y \vee x = (y \vee x) \wedge y$ and the lemma, we have $(y \vee x) \vee y = y$, as was to be shown.

Related concepts

• lattice

• band

References

• Wikipedia, Skew lattice