“Lattice” can be defined in two “equivalent” ways :
1. From POSET(Relation) point of view
2. From Algebra point of view.
Both definitions are equivalent in the sense that one definition implies another.
We already know Lattices as Posets. Let us see Lattices as Algebraic Structures.
Lattice as algebraic structures :
A lattice can also be defined as an algebra $(L,∧,∨)$ on a set L with two binary operations ∧ (meet) and ∨ (join). The algebra satisfies the following identities :
- Commutativity: $a∧b=b∧a; a∨b=b∨a$.
- Associativity: $(a∧b)∧c=a∧(b∧c); (a∨b)∨c=a∨(b∨c)$.
- Idempotence: $a∧a=a; a∨a=a$.
- Absorption: $a∧(a∨b)=a; a∨(a∧b)=a$,
where $a,b,c∈L$.
Basically, An algebraic structure $(L,∧,∨)$, consisting of a set $L$ and two binary operations $(∧,∨)$ , on $L$ is a lattice if “Commutative, Associative, Absorption, Idempotence” Properties are satisfied.
(Also NOTE that “Closure” property is implicit in the definition of “binary operation”, So, it need not be mentioned explicitly that it must be satisfied ; Also the idempotence property is redundant as it is implied by other properties)
Now, Coming to the given question, The given set with two binary operations LCM, GCD ; satisfies ALL the mentioned properties, So, it is a Lattice.
https://www.math24.net/lattices
https://en.wikipedia.org/wiki/Lattice_(order)
https://mathworld.wolfram.com/Lattice.html
https://ncatlab.org/nlab/show/lattice
https://math.stackexchange.com/questions/1158018/definition-of-algebraic-structure