logic – Modeling Equality in an ILP

Let's say we have a variable $ a = mathbb {Z} ^ n $ and another $ M = {0,1 } ^ {n times L} $. I want to model the logic $$ M_ {i, j} iff (a_i == j) $$

I have two cases to consider, $ a_i leq i $ and $ a_i leq L $.

Option 1

We can use the method described in the answer here by modeling the difference of $ a_i $ and $ j. So we have $$ M_ {i, j} ssi a_i-j == 0 $$

That implies $ 1 $ additional binary variable as well as $ 4 $ potentially big "Big M" constraints.

Option 2

We can break down $ a_i $ in its bits using $ log L $ additional binary variables. We can also calculate the corresponding binary bit failures for each $ 0 leq j leq L $. Let $ bit ^ {(a)} _ {i, k} $ Be the $ kthe little $ a_i $ and let $ bit_ {j, k} ^ * $ Be the $ kthe bit of the integer $ j .
$$ a_i = sum_ {k = 0} ^ { log L} 2 ^ kbit ^ {(a)} _ {i, k} $$

We now have the following logic:
$$ M_ {i, j} sif the bit ^ {(a)} _ {i, k} odot the bit {j, k} ^ *
; ; ; forall k ; ; ; ; 0 leq k leq log L $$

$$ M_ {i, j} = bigwedge_ {k = 0} ^ { log L} bit ^ {(a)} _ {i, k} odot bit_ {j, k} ^ * $$

we can use this publication for NXOR pairs of variables, storing their results in another variable $ z = {0,1 } ^ { log L} $. We can AND these together using the constraints $$ M_ {i, j}> = sum z_i $$ $$ M_ {i, j} leq z_i ; ; ; forall i $$

If I'm option 2, we'll have $ log L $ new variables and 1 constraint to represent the binary decomposition of $ a_i $. We then $ log L $ XNOR variables, each of which can be represented with 4 constraints. We can finally represent $ M_ {i, j} $ with $ log L + 1 $ more constraints. Therefore, we have a total of $ 5 log L $ constraints and $ 2 log L $ new variables.

TLDR)

  1. Is there a way to represent a Boolean variable for the equality of 2 positive integer integers in an ILP that do not involve Big M constraints or binary failures, and if not, which one is generally used?

Note* This is a repost, however I have republished it with only one of the initial questions because of @DW's suggestion and response.