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 antisymmetric. 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:
$$\text{xymatrix}\mathrm{\&}a\vee b\text{ar}\mathrm{@}[d]\mathrm{\&}\mathrm{\&}c\text{ar}\mathrm{@}[ld]\text{ar}\mathrm{@}[rd]\mathrm{\&}a\text{ar}\mathrm{@}[rd]\mathrm{\&}\mathrm{\&}b\text{ar}\mathrm{@}[ld]\mathrm{\&}a\wedge b\mathrm{\&}$$ 
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 meetassociative latticoid iff ${L}^{*}$ is joinassociative.
Title  latticoid 

Canonical name  Latticoid 
Date of creation  20130322 16:31:02 
Last modified on  20130322 16:31:02 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06F99 