For an application I'm working on, I'm interested in using a field that has all the properties of the finite Galois field GF(2). For reference, the addition operation $+$ in GF(2) is defined by:
$$ \begin{array}{c c|c} A & B & A+B \\\hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} $$
which in digital logic is referred to as exclusive-OR (XOR) and is sometimes denoted $A \oplus B$. The appeal of using a field such as GF(2) is that all the standard results from linear algebra can be applied to vectors and matrices defined over this field.
Is there a comparable algebraic structure that exists with all of the same operations defined on its elements (of which there are also only two-- $0$ and $1$) except that $+$ is defined as the OR operation rather than XOR? In other words, the truth table would be given by:
$$ \begin{array}{c c|c} A & B & A+B \\\hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \end{array} $$ If such a structure exists, would it also be considered a field? Will it posses the "usual" properties such that it can be used in operations in a vector space?
(I acknowledge that this question was answered in the comments; I am posting as an answer here for completeness.)
Such a structure exists by the simple fact that you have defined it, but it is not a field.
Every field has an additive inverse. In this case that means that $0+x=0$ and $1+x=0$ must both have solutions. This can be checked in your table. The first equation has a solution (in fact, your sum has an additive identity). But inspection of the table shows that the second equation does not have a solution. Since there is no additive inverse, the structure you propose is not a field.