# 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

## 1 Answer

One of the problems with removing elements from a structure is that these elements may be images of other elements under the basic operations. In some lattices, zero can be a meet of nonzero elements, for example.

One way to cope with this is to adopt the notion of partial algebra, where a basic operation can be a function that is not defined on some inputs. For more on this, I refer you to George Graetzer's Universal Algebra text.

A second way to cope with this is to find a way to "bring back" the excised elements. In tame congruence theory, one considers restrictions of an algebraic structure to a subset which is not a subalgebra, and the subset may not be closed under the basic operations. For the goal of the study, an auxiliary polynomial function ( built out of basic operations and additional constants as needed) p is used which maps the subset into itself, and now this subset along with p and the original enclosing structure is studied. If you want to dive into that theory, I refer you to the Structure of Finite Algebras by Hobby and McKenzie. Since then, summaries of the theory have been made available. I suspect Ross Willard has written one, but I did not remember the title.

It is curious to me why you would want to remove elements to "remove identities". Except for the closure aspect, the subset will obey the same set of identities and possibly more. Adding elements is a way of breaking identities. Perhaps if you gave a little more of your motivation and your thoughts on why removing elements helps, we can tell you why it doesn't and suggest something that moves you toward your desired goal.

Gerhard "More Than One Piece Missing" Paseman, 2019.01.21.