I am trying to derive some basic relations for the height and width of the direct product and the coproduct of posets. I feel that these are very basic and should be written somewhere, however, I cannot find a reference.

Short question is: is there a short expression for the following quantities, representing height and width of product and coproduct of posets? And do they hold also in the case of infinite cardinality?

**Edit: current status (with help from Harry Altman, David Spivak) of this question:**

**[resolved]** $\color{green}{ w(P\coprod Q) = w(P)+w(Q) }$

**[resolved]** $\color{green}{ h(P\coprod Q) = \max\{h(P), h(Q)\}}$

**[resolved]** Assuming $P$ and $Q$ not empty, then $\color{green} {h(P \times Q) = h(P)+h(Q)−1 }$. (For empty posets, then $h(P \times Q) = 0 \neq h(P)+h(Q)-1$.)

**[resolved]** From a theorem in Berzukov, Roberts, "On antichains in product posets", it follows that the width can be bounded as follows, with both bounds attainable:

$\color{green}{ w(P)w(Q)\leq w(P\times Q) \leq \min\{|P|\ w(Q), |Q|\ w(P)\}} $

Original question below.

**Preliminaries** Define:

- $C_n$ to be a chain of size $n$. For example take $C_n = \langle\{1, \dots, n\}, \leq\rangle$.
- $A_n$ to be an antichain of size $n$, that is, a set with $n$ incomparable elements.
- $P \times Q$ the direct product of two posets.
- $G_{m,n}$ is a grid; for example $G_{m,n} = C_n \times C_m$.

The height and width of a poset are defined as:

- the height $h(P)$ is the cardinality of the longest chain in $P$.
- the width $w(P)$ is the cardinality of the longest antichain in $P$.

Some simple examples:

Width of a **chain**: $w(C_n) = 1$.

Height of a chain: $h(C_n) = n$.

Width of an **antichain**: $w(A_n) = n$.

Height of a antichain: $w(A_n) = 1$.

Width of an $m\times n$ **grid**: $w(G_{m\times n}) = \min\{m,n\}$

Height of an $m\times n$ grid: $w(G_{m\times n}) = m + n -1$

**Questions**

Is there a simple expression for the height and width of a product and a coproduct of a poset?

This is what I got so far.

For a *co-product*:

The height must be the maximum of the two heights, because chains belonging to different factors are uncomparable:

$ h( P \coprod Q) = \max\{ h(P), h(Q) \}$

For the width, the widths of the factors sum together:

$ w( P \coprod Q) = h(P) + h(Q) $

This is because I can take an antichain $S_1$ in $P$ and one antichain $S_2$ in Q, and then $S_1\cup S_2$ is an antichain in $P \coprod Q$.

For a *product*, I am not sure.

For the height of a product I can certainly say that

$h(P\times Q) \geq h(P) + h(Q) - 1$

because I can construct a chain of that size. If $C=\{1,2,\dots,h(P)\}$ is the longest chain in $P$ and $D = \{a,b,\dots\}$ the longest chain in $Q$ then I can construct the chain $E = \{(1,a), (2,a), \dots, (h(P), a), (h(P), b), \dots\}$ that has height $ h(P) + h(Q) - 1$.

I am also not sure if any of the above fails for posets of infinite cardinality.