# What is the method to relax or weakening a structure “ripping off” from it all its identity elements?

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?

• You mean like en.wikipedia.org/wiki/Semigroup , for instance—a general definition of a given structure without the requirement for identity element(s)? Or do you mean given an actual algebraic structure, remove the identity element(s) and then still think about that structure? You can take a lattice, remove its top and bottom, and then the resulting set still has at least partially-defined meet and join. – David Roberts Jan 21 '19 at 19:40
• As an example, in the method of forcing in set theory, people often consider a complete Boolean algebra $B$, as a partially-ordered set, and then remove the bottom element $\bot$ to get the partially ordered set $B\setminus \bot$ and then use that. – David Roberts Jan 21 '19 at 19:42