I thought of viewing $\bar{\mathbb{R}}$ as a relative subalgebra of a total algebra $\mathbb{R}^\ast := \bar{\mathbb{R}} \cup \{\ast\}$, wherein an output of an operation, if not already defined in $\bar{\mathbb{R}}$, is set equal to $\ast$. In a sense, $\ast$ acts as the definition for those extended real sums or products that were previously undefined, while also absorbing everything it touches algebraically. In this construction $\ast$ is not equal to any extended real number.
I'm not claiming this is an original idea. I think this is a typical approach of viewing a partial algebra as a relative subalgebra of a total algebra. Nonetheless, I liked it. Before I get into why, my question is, does the total algebra I'm calling $\mathbb{R}^\ast$ have a name?
Definitions... Addition in $\bar{\mathbb{R}}$ is defined for all extended real pairs except $(\pm\infty,\mp\infty)$. By keeping these definitions, and further defining $\pm\infty + \mp\infty = \ast$ and also $a+b = \ast$ whenever $\ast \in \{a,b\}$, we obtain the definition of addition on $\mathbb{R}^\ast$. Similarly, multiplication in $\bar{\mathbb{R}}$ is defined for all extended real pairs (I let $\pm\infty\cdot 0 = 0$), so by further defining $ab = \ast$ whenever $\ast \in \{a,b\}$ we get the definition of multiplication in $\mathbb{R}^\ast$. To extend negation we just let $- \ast = \ast$.
I even went so far as to give $\ast$ the name "undefined", so I could use the phrases "$a$ is defined" and "$a$ is undefined" to respectively mean that $a \neq \ast$ and $a = \ast$. For instance, to say the sum $a+b$ is undefined is to say $a+b = \ast$.
It turns out that addition and multiplication in $\mathbb{R}^\ast$ are associative and commutative, $0$ and $1$ act as additive and multiplicative identities, respectively, and negation is an involution. Moreover, $\mathbb{R}^\ast$ is real-cancellative in the sense that $x +(-x) = 0$ only holds for real $x$.
Distributivity is a little more delicate. We'll say that a sum $b+c$ is heterogeneous in case $$(b,c) \;\; \in \;\; [-\infty,0)\times(0,+\infty] \;\;\; \cup \;\;\;(0,+\infty] \times [-\infty,0). $$ So a defined heterogeneous sum is one that is heterogeneous and equal to an extended real number, whereas an undefined heterogeneous sum is one that is heterogeneous and equal to $\ast$. For any $a,b,c \in \mathbb{R}^\ast$, $$a(b+c) = ab + ac$$ so long as $a \neq \pm\infty$ when $b+c$ is a defined heterogeneous sum, and $a \neq 0$ when $b+c$ is an undefined heterogeneous sum.
Why I like it... I know there is nothing special about $\mathbb{R}^\ast$. I'm just using new symbols to say old things. I guess this is the way math has always gone. But it does make it easier for me. I'll only ever care about algebraic expressions whose terms are in $\bar{\mathbb{R}}$, but it's nice to know I don't have to worry about checking for infinities in said expressions - if they don't evaluate to an extended real number, they'll at least evaluate to something ($\ast$).
There is similar ease in considering algebraic expressions involving extended real functions on some domain $X$. For extended real functions $e,f,g,h$, it may be that $d = f(g+h) + e$ is not extended-real for some $x$, but that is no problem; $d(x) = \ast$ for such $x$, and $d$ at least exists in some setting. You can also use indicator-like functions $$\ast_E(x) = \begin{cases}\ast, & x \in E, \\ 0, & x \notin E \end{cases}$$ to adjust expressions, and even call these functions "adjustors". For instance, for any functions $f,g,h : X \to \mathbb{R}^\ast$, $$h(f+g) + \ast_A = hf+hg + \ast_B$$ and $$f + (-f) = \ast_C$$ where $$A = \{x \in X: \,h(x) = \pm\infty \;\text{and} \; f(x) + g(x) \; \text{defined heterogenous sum}\} $$ $$B = \{x \in X: \,h(x) =0 \;\text{and} \; f(x) + g(x) \; \text{undefined heterogenous sum}\}$$ $$C = \{x \in X \;: \; f(x) \; \text{not real}\}.$$ Thus you could say things like $\mathbb{R}^\ast$ function spaces are adjusted-cancellative and adjusted-distributive.
But I ask again, does $\mathbb{R}^\ast$ have a name? Visually, I picture $\mathbb{R}^\ast$ a coproduct of topological spaces: a closed line segment with $\ast$ hovering off to the side like a satellite. Is there anything like this construction for the complex numbers? Whereby the complex numbers are visualized as an open circle, and to close that circle a pair of antipodal "infinities" are defined as the endpoints of each diameter through the circle, so that each such infinity is distinct; and then hovering off to the side of the closed circle is the single point $\ast$. You could call this space $\mathbb{C}^\ast$ if you're comfortable doing so.
It looks like you invented Wheel theory.
In that algebraic system, $0/0=\bot$ is called "nullity". It looks like all your rules are satisfied.
Particularly, $\infty-\infty=\bot$, $0\bot=\bot$.