Exercise 13, § 2

(a) Let $\mathrm{Is}(\Gamma, \Gamma')$ be the relation "$\Gamma$ is an ordering (on E), and $\Gamma'$ is an ordering (on E'), and there exists an isomorphism of E, ordered by $\Gamma$, onto E', ordered by $\Gamma'$". Show that $\mathrm{Is}(\Gamma, \Gamma')$ is an equivalence relation on every set whose elements are orderings. The term $\tau_\Delta(\mathrm{Is}(\Gamma, \Delta))$ is an ordering called the order-type of $\Gamma$ and is denoted by Ord $(\Gamma)$, or Ord(E) by abuse of notation. Two ordered sets are isomorphic if and only if their order-types are equal.

(b) Let $R\{\lambda, \mu\}$ be the relation : "$\lambda$ is an order-type, and $\mu$ is an order-type, and there exists an isomorphism of the set ordered by $\lambda$ onto a subset of the set ordered by $\mu$". Show that $R\{\lambda, \mu\}$ is a preorder relation between $\lambda$ and $\mu$. It will be denoted by $\lambda \prec \mu$.

(c) Let I be an ordered set and let $(\lambda_\iota)_{\iota \in I}$ be a family of order-types indexed by I. The order-type of the ordinal sum (§ 1, Exercise 3) of the family of sets ordered by the $\lambda_\iota$ ($\iota \in I$) is called the ordinal sum of the order-types $\lambda_\iota$ ($\iota \in I$) and is denoted by $\sum\limits_{\iota \in I} \lambda_\iota$. If $(E_\iota)_{\iota \in I}$ is a family of ordered sets, the order-type of $\sum\limits_{\iota \in I} E_\iota$ is $\sum\limits_{\iota \in I} \mathrm{Ord}(E_\iota)$. If I is the ordinal sum of a family $(J_\varkappa)_{\varkappa \in K}$, show that

(d) Let I be a well-ordered set and $(\lambda_\iota)_{\iota \in I}$ a family of order-types indexed by I. The order-type of the lexicographic product of the family of sets ordered by the $\lambda_\iota$ ($\iota \in I$) is called the ordinal product of the family $(\lambda_\iota)_{\iota \in I}$ and is denoted by $\mathop{\mathsf{P}}\limits_{\iota \in I} \lambda_\iota$. If $(E_\iota)_{\iota \in I}$ is a family of ordered sets, the order-type of the lexicographic product of the family $(E_\iota)_{\iota \in I}$ is $\mathop{\mathsf{P}}\limits_{\iota \in I} \mathrm{Ord}(E_\iota)$. If I is the ordinal sum of a family of well-ordered sets $(J_\varkappa)_{\varkappa \in K}$, indexed by a well-ordered set K, show that

(e) We denote by $\lambda + \mu$ (resp. $\mu\lambda$) the ordinal sum (resp. ordinal product) of the family $(\xi_\iota)_{\iota \in J}$, where $J = \{\alpha, \beta\}$ is a set with two distinct elements, ordered by the relation whose graph is $\{(\alpha, \alpha), (\alpha, \beta), (\beta, \beta)\}$, and where $\xi_\alpha = \lambda$, $\xi_\beta = \mu$. Show that if I is a well-ordered set of order-type $\lambda$, and if $(\mu_\iota)_{\iota \in I}$ is a family of order-types such that $\mu_\iota = \mu$ for each $\iota \in I$, then $\sum\limits_{\iota \in I} \mu_\iota = \mu\lambda$. We have $(\lambda + \mu) + \nu = \lambda + (\mu + \nu)$, $(\lambda\mu)\nu = \lambda(\mu\nu)$, and $\lambda(\mu + \nu) = \lambda\mu + \lambda\nu$ (but in general $\lambda + \mu \neq \mu + \lambda$, $\lambda\mu \neq \mu\lambda$, $(\lambda + \mu)\nu \neq \lambda\nu + \mu\nu$).

(f) Let $(\lambda_\iota)_{\iota \in I}$ and $(\mu_\iota)_{\iota \in I}$ be two families of order-types indexed by the same ordered set I. Show that, if $\lambda_\iota \prec \mu_\iota$ for each $\iota \in I$, then $\sum\limits_{\iota \in I} \lambda_\iota \prec \sum\limits_{\iota \in I} \mu_\iota$ and (if I is well-ordered) $\mathop{\mathsf{P}}\limits_{\iota \in I} \lambda_\iota \prec \mathop{\mathsf{P}}\limits_{\iota \in I} \mu_\iota$. If J is a subset of I, show that $\sum\limits_{\iota \in J} \lambda_\iota \prec \sum\limits_{\iota \in I} \lambda_\iota$ and (if I is well-ordered and the $\lambda_\iota$ are non-empty) $\mathop{\mathsf{P}}\limits_{\iota \in J} \lambda_\iota \prec \mathop{\mathsf{P}}\limits_{\iota \in I} \lambda_\iota$.

(g) Let $\lambda^*$ denote the order-type of the set ordered by the opposite of the ordering $\lambda$. Then we have

where $I^*$ denotes the set $I$ endowed with the opposite of the ordering given on $I$.