In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them
For example, we know that lattice is an algebraic structure, with the binary operations $\wedge$ and $\vee$ and the constants $\top$ and $\bot$. (These correspond, respectively, to binary and nullary meets and joins in the poset-theoretic definition; accordingly, they are read ‘meet’, ‘join’, ‘top’, and ‘bottom’.) Here are the axioms for these operations:
$\wedge$ and $\vee$ are each idempotent, commutative, and associative;
the absorption laws: $a \vee (a \wedge b) = a$, and $a \wedge (a \vee b) = a$;
$\top$ and $\bot$ are the respective identities of $\wedge$ and $\vee$.
I want to 'remove' identities of ∧ and ∨ like a sort of de-identity of $\wedge$ and $\vee$ operations to have a de-identity operations.
I'm interested in being able to insert them in a second moment because I want a structure without operations identities not without operations!, or, in general, a structure without identities but not without a structure!
Is there a method to relax or weaken a structure "ripping off" from it all its identity elements that are present respect to a binary operation on that set?