Cases where ANY 2 of 3 +/- choices select one of four possible elements

It seems to me you can consider two of the properties to be the "encoding" of the base and treat the third property as a "parity check bit." You can make an arbitrary choice of which property to designate as the "parity check bit." I'm not sure if this qualifies as a "mathematical structure," however.

A possible difficulty in trying to discuss a mathematical structure for your observation about the bases of DNA or RNA is that the notation you have used is really not mathematical. For example, in normal usage an equation such as "$a = +$" is nonsense, and writing "$a = + \text{ or } -$" is no better. Also if $Y,$ $W,$ and $K$ are mathematical objects to which the operators $+$ or $-$ can be applied, how do we know that $+Y \neq +W$ or that $+Y \neq -K$ (among many other possibilities)?

What you could do instead is define functions named $Y,$ $W,$ and $K,$ each of which maps any base to either $1$ or $-1$ depending on whether the base has the property corresponding to that function's name or the opposite property. For example, $Y(t) = Y(c) = 1$ and $Y(a) = Y(g) = -1.$

Now that we have some actual mathematical definitions associated with the bases, we can say that for any arbitrary base $x$ in either DNA or RNA, $$ Y(x) \times W(x) \times K(x) = 1, $$ which is a constraint on the properties of the base. Using this constraint, it is clear that whenever you know two of the properties of the base, you can find the third property by solving this equation algebraically. Given that the three properties uniquely identify one of the bases of DNA (or one of the bases of RNA), this constraint tells us that any two of the properties are sufficient to identify the base.

This is not the only possible formulation, however. You could just as well use $1$ and $0$ as the indicators of "has the property" or "has the opposite property", so that $Y(t) = Y(c) = 1$ and $Y(a) = Y(g) = 0.$ Then the constraint would be $$ Y(x) + W(x) + K(x) \equiv 1 \pmod 2, $$ or in more ordinary language, the sum of the three function values must be an odd number. This is closer to the way "parity bits" are interpreted in telecommunication and computing, but the effect you are concerned with is the same, namely, by knowing any two of the three properties we can find the third.