I say that a poset is lower-bounded (resp., upper-bounded) if it has a minimum (resp., maximum) element, and I say that the poset is bounded if it is both lower and upper-bounded. In algebra, sometimes one is interested in the elements of a certain lower and/or upper-bounded poset that are just “one step” above/below the minimum/maximum element. For example, an element is a ring $a\in R$ can be said to be irreducible if the ideal $(a)$, inside the poset of principal ideals, is just “one step” below $(1)$. That is, if $(a)\neq(1)$ and it does not exist $b\in R$ with $$ (a)\subsetneq(b)\subsetneq(1). $$ Another example are maximal ideals: they are the analogous thing in the poset of all ideals of a ring.
An example of the dual behavior would be the height 1 prime ideals in the poset of prime ideals of an integral domain.
So my question is: do you know some mathematical terminology to call these elements in a lower or upper-bounded poset? (Or just a name for the generalized notion in a poset without maximum or minimum.) Of course “height 1” and “coheight 1” would be terminological candidates (the (co)height of a prime ideal is defined in all rings, not only for integral domains). However, my background is mostly algebra (where the “(co)height” terminology comes from), so my main curiosity is to see if other fields of math have a different name for this phenomenon (and maybe looking for a word that is not numbered, as in “(co)height $n$”).
Also, the word “(co)height” can be confusing to use to refer to this phenomenon in a poset of non-prime ideals of a ring, because the (co)height is defined in this way for non-prime ideals.
I think you mean atoms and co-atoms. The wiki article includes a reference to Introduction to Lattices and Order, by Priestley.