One main effect and one interaction in R using multiple regression, is that possible? And why am I getting two interaction terms in output?

I have two factors that are fully crossed, the levels of the factor are each coded 0 and 1. I am running a regression testing for one main effect and one interaction. The following is my logistic regression formula:

m1=glmer(y~1+A+A:B+(1|Participants)+(1|Word),data=data, family = "binomial")


I am wondering if this is acceptable (only testing for one main effect and an interaction), and also why I am getting two interaction terms in my output:

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.18740    0.21600  -0.868  0.38561
A1           0.74546    0.28399   2.625  0.00867 **
A0:B1        0.01537    0.28244   0.054  0.95662
A1:B1        0.15884    0.28650   0.554  0.57929

• Please migrate it to CV. – user227710 Jul 3 '15 at 0:17
• Is there some reason why you are not including the main effect of B in your model? It's generally best to include main effects of variables for which you are examining interactions. – EdM Jul 3 '15 at 15:22

There are 4 possible states for the interaction of A and B. That is
1. A0:B0, and not shown as it is absorbed into the intercept term for the regression.
2. A1 (implicitly A1:B0).
3. A0:B1.
4. A1:B1.