latticoid

A latticoid is a set $L$ with two binary operations, the meet $\wedge$ and the join $\vee$ on $L$ satisfying the following conditions:

1. (idempotence) $x\wedge x=x\vee x=x$ for any $x\in L$,

2. (commutativity) $x\wedge y=y\wedge x$ and $x\vee y=y\vee x$ for any $x,y\in L$, and

3. (absorption) $x\vee (y\wedge x)=x\wedge (y\vee x)=x$ for any $x,y\in L$.

A latticoid is like a lattice without the associativity assumption, i.e., a lattice is a latticoid that is both meet associative and join associative.

If one of the binary operations is associative, say $\wedge$ is associative, we may define a latticoid as a poset as follows:

$$x\le y\text{ iff }x\wedge y=x.$$

Clearly, $\le$ is reflexive, as $x\wedge x=x$. If $x\le y$ and $y\le x$, then $x=x\wedge y=y\wedge x=y$, so $\le$ is anti-symmetric. Finally, suppose $x\le y$ and $y\le z$, then $x\wedge z=(x\wedge y)\wedge z=x\wedge (y\wedge z)=x\wedge y=x$, or $x\le z$, $\le$ is transitive.

Once a latticoid is a poset, we may easily visualize it by a diagram (Hasse diagram), much like that of a lattice. Position $y$ above $x$ if $x\le y$ and connect a line segment between $x$ and $y$. The following is the diagram of a latticoid that is meet associative but not join associative:

It is not join associative because $(a\vee b)\vee c=a\vee b$, whereas $a\vee (b\vee c)=a\vee c=c\ne a\vee b$.

Given a latticoid $L$, we can define a dual $L^{*}$ of $L$ by using the same underlying set, and define the meet of $a$ and $b$ in $L^{*}$ as the join of $a$ and $b$ in $L$, and the join of $a$ and $b$ (in $L^{*}$) as the meet of $a$ and $b$ in $L$. $L$ is a meet-associative latticoid iff $L^{*}$ is join-associative.