I am looking for a reference on well-founded ordered sets, that would mention the following notions and results:
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a) Consider a well-ordered set $(I,<)$ and a family $(X_i,<_i)$ of well-founded ordered sets. The sum $\sum \limits_{i \in I} X_i$ of the family is the set $\bigcup \limits_{i \in I} X_i \times \{i\}$ ordered by $(x,i)<(y,j)$ if $i<j$ or $i=j$ and $x<_iy$.
b) Given a well-founded ordered set $(X,<)$ and $x \in X$, the rank of $x$ is the ordinal $r(x)=\sup \{r(y)+1 : \ y<x\}$ and the rank of $X$ is the supremum of $r(X)$.
c) For ordinals $\alpha,\beta$, the ordinal $\alpha +\beta$ is the order type of $\sum \limits_{i \in I} X_i$ where $I=2$ and $(X_0,X_1)=(\alpha,\beta)$. The ordinal $\alpha \beta$ is the order type of $\sum \limits_{i \in I} X_i$ where $I=\beta$ and $X_i=\alpha$ for all $i\in \beta$.
d) The rank of $\sum \limits_{i \in I} X_i$ is the order type of $\sum \limits_{i \in I} r(X_i)$.
Does someone know a reference with all of that or at least a), b) and d)?