I don't know much about singular cardinal combinatorics, so I apologize in advance if I write something that is wrong or looks funny. First let me recall some basic definitions.

Let $\lambda$ be a singular cardinal and let $(\kappa_i : i < \text{cf}(\lambda))$ be a sequence of regular cardinals cofinal in $\lambda$.
A sequence $(f_\alpha : \alpha < \lambda^+)$ of functions $f_\alpha \in \prod_{i < \text{cf}(\lambda)} \kappa_i$ is called a *scale* if it is:

(1) increasing modulo $<^*$, where $f <^* g$ means that $f(i) < g(i)$ for all but a bounded set of $i < \text{cf}(\lambda)$, and

(2) cofinal in $\prod_{i < \text{cf}(\lambda)} \kappa_i$ modulo $<^*$.

A scale is said to be *good* if it satisfies the additional property

(3) for club many $\alpha < \lambda^+$, if $\text{cf}(\alpha) > \text{cf}(\lambda)$ then $\alpha$ is a *good point*, meaning that there is an unbounded set $S \subset \alpha$ and an ordinal $i < \text{cf}(\lambda)$ such that the sequence $(f_\alpha \restriction [i,\text{cf}(\lambda)): \alpha \in S)$ is pointwise increasing.

The proof of the fact "if $\lambda$ is a singular cardinal and there is a strongly compact cardinal between $\text{cf}(\lambda)$ and $\lambda$, then there is no good scale of length $\lambda^+$" seems to use only properties (1) and (3) but not property (2). So my questions are:

(A) Do sequences of length $\lambda^+$ satisfying (1) and (3) have a name (e.g. "good pseudo-scale")?

(B) Is their existence equivalent to the existence of good scales?

solast year! ;-)) $\endgroup$