1. ## V_gp vocabulary

if we let that V_gp be the vocabulary {+,0} where + is abinary function and 0 is a constant . we use the notation x+y to denote the term +(x,y) . Let gama be the conjunction of the following V-sentence: for all x the exist y for all z (x+(y+z)=(x+y)+z)
for all x (( x+0=x) conjunction ( o+x=x))
for all x ( there exists y (x+y=0) conjunction the exist z (z+x=0))
how can I give a model of gama that does not model the sentence for all x for all y((x+y)=(y+x)).

2. You can take any non-commutative group as a model. One examples is invertible (n x n)-matrices under multiplication (thus, + is interpreted as multiplication and 0 is interpreted as the identity matrix). Another example is the group of permutations $S_3$, which is the group of bijections of {1, 2, 3} under composition. It is also a group of isometries (transformations that preserve distances, such as rotations, shifts and reflections) that map an equilateral triangle into itself.

It's not clear to me why there is an existential quantifier in
for all x the exist y for all z (x+(y+z)=(x+y)+z)
It does not seem more difficult to find a model of a similar sentence where all quantifiers are universal.

3. so , as I understood , I can say that that nXn matrices under multiplication models the first three sentences but doesnot model for all x for all y((x+y)=(y+x)).

4. Yes, + is interpreted as matrix multiplication, which is non-commutative in general.