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



A "good scale" that is not really a scale
Your pseudo-good scale is indeed equivalent to a good scale, although the good scale possibly lives on a different product. This follows easily from the construction of a scale given in Cummings' Notes on Singular Cardinal Combinatorics.


A basic theorem of Shelah: If a $<^\*$-increasing sequence of functions has stationary many good points of cofinality $\kappa$ for some $\kappa<\lambda$, then it has an exact upper bound $g$ such that $\mathrm{cf}(g(i))>\kappa$ for but boundedly many $i$. 


Taking such an exact upper bound, you can thin to $X\subseteq \mathrm{cf}(\lambda)$ so that $i\mapsto \mathrm{cf}(g(i)): i\in X$ is increasing and unbounded in $\lambda$. Then you can fix cofinal subsets of $g(i)$ of order-type $\mathrm{cf}(g(i))$ for each $i\in X$ and modify the functions to live on $\prod\_{i\in X} \mathrm{cf}(g(i))$. You can check that this preserves (3), and the resulting sequence of functions is cofinal in the new product.


