Definition of frame in order theory I came across the definition of frame in a lecture as follows :

Definition (frame). A frame is a poset $(L, \le)$ which has finite meets and all joins, and which satisfies the following infinite distributive law, where $S$ is an arbitrary subset of $L$:
$$a \wedge \bigvee S = \bigvee \{a \wedge s ~|~s \in S\}.$$

In my understanding, a poset $L$ having "all joins" means that any of its subset $S$ has a join $\bigvee S \in L$. However, a poset having "all joins" has "all meets", and a frame is, therefore, a particular complete lattice.
Consequently, I'm not sure what means "has finite meets" as a frame has "all meets". I have already consulted the definition in the entry of nlab (frame) and the book of Johnstone on Stone spaces [1] which I recall here:

Definition in nlab (frame). A frame $\mathscr{O}$

*

*is a poset

*that has

*

*all small coproducts, called joins ⋁

*all finite limits, called meets ∧



*and which satisfies the infinite distributive law.




Definition in [1] (The category Frm). The category Frm is the category whose objects are complete lattices satisfying the infinite distributive law, and whose morphisms are functions preserving finite meets and arbitrary joins.

Nonetheless, I'm not familiar with category theory at all and I'm looking for a purely order-theoretic definition of frame.
EDIT:
There is a similar question about frame. If I understand correctly, a frame is merely a complete lattice satisfying the infinite distributive law (if we don't consider morphism) ?
[1] Johnstone, Peter T., Stone spaces, Cambridge Studies in Advanced Mathematics, 3. Cambridge etc.: Cambridge University Press. XXI, 370 p. (1986). ZBL0586.54001.
 A: There's a subtle difference between what I'll call primary structure and secondary structure. By "primary structure" I mean those properties explicitly given in the definition, while by "secondary structure" I mean those further properties which we can derive from the primary structure.
Some of the time this isn't an important distinction - e.g. in (classical) model theory for the most part - but other times it's quite important. Most obviously, it impacts the relevant notions of substructure and homomorphism:


*

*For $A$ to be a subframe of $B$, we need that $(i)$ $A$ has all joins and finite meets and $(ii)$ those agree with those in $B$. But we can have a subframe $A$ of $B$ and an infinite set $X\subseteq A$ such that the greatest lower bound of $X$ in the sense of $A$ is strictly below the greatest lower bound of $X$ in the sense of $B$: agreement on the level of the "secondary" structure isn't part of the definition of substructure.

*Similarly, a homomorphism of frames needs to preserve finite meets and all joins, but does not need to preserve infinite meets: we can have a frame homomorphism $f:A\rightarrow B$ and an infinite $X\subseteq A$ with greatest lower bound $a$ in the sense of $A$ such that the greatest lower bound of $f[X]$ in the sense of $B$ is strictly above $f(a)$.
(This is of course redundant since subframes are examples of frame homomorphisms, but it still "feels" right to me to list them both; dunno why.)
Note that this issue appears, not at the level of the individual structures, but "one level higher" when we talk about how the relevant structures interact.
