The mathematical use of "lattice" is unfortunate, since it has other nonmathematical usages. Gardners use wooden lattices for climbing vines. And particle physicists may impose a grid lattice upon interaction-space to study particle properties. Confusing! But the mathematical use is standard, and must be suffered. (I didn't did it!)

I'll sketch the formal definition of this concept. Then, in case you need this, I'll give proto-education, with the aid of BNF (Backus Naur Form) language, which displays informal language on the left and formal language on the right":

• «informal terms» := «formal terms».

A lattice is a partial ordering such that each pair of elements has a least upper bound (a.k.a supremum, minmax) and a greatest lower bound (a.k.a. infimum, maxmin), where an ordering if partial if it is also reflexive and antisymmetric.

Since any ordering relation must be transitive, a partial ordering (a.k.a poset) is an ordering that is also reflexive and antisymmetric. (The best known formal partial orderings are "less than or equal to" (£) and "greater than or equal to" (³.)

You are proto-educated in partial ordering IF YOU UNDERSTAND INCLUSION RELATIONS. You are proto-educated in reflexivity in Grammar. In a reflexive sentence, the subject and direct object are identical. For example, "I massage myself."

BNF

«relation» := R
«first party related to second party» := «first»R«second».
«if statement A, then statement B» := A É B
«and» := &
«transitive relation» := ((A R B) & (B R C)) É (A R C)
«reflexive relation» := A R A
«antisymmetric relation» := ((D & E) & (E R D)) É (D = E)
«or» := OR
«connected relation» := (x R y) OR (y R x)

"less than or equal to" (£) and "greater than or equal to" (³) as PARTIAL ORDERING

x £ x    &    x ³ x [reflexivity]
((x £ y) & (y £ x)) É (x = y)|| ((x ³ y) & (y ³ x)) É (x = y) [antisymmetry]
((x £ y) & (y £ z)) É (x £ z)|| ((x ³ y) & (y ³ z)) É (x É z) [transitivity]

The "partial" means that this relation is NOT CONNECTED -- that is, RELATANDS EXIST IN THE SYSTEM such that NEITHER DOMINATES THE OTHER.

Some Examples of PARTIAL ORDERING

• included in
• is contained in
• is dominated by
• is pecked by
• subset
• CONDITIONAL: "if _, then _"
• factor of
• subordinate

As stated above a lattice is a partial ordering such that each pair of elements has a least upper bound (a.k.a. supremum, minmax) and a greatest lower bound (a.k.a. infinmum, maxmin).