In section 2.5.1 of Federer's book "Geometric Measure Theory," given a set $X$, he says "By a lattice of functions on $X$ we mean a set $L$ whose elements are functions mapping $X$ into $\mathbb{R}$, and which satisfies the following condition: if $0\leq c<\infty$, and if $f$ and $g$ belong to $L$, so do $f+g$, $cf$, $\inf\{f,g\}$ and $\inf\{f,c\}$; in case $f\leq g$, also $g-f$ belongs to $L$."
Is there standard terminology for this object? It seems to be quite close but not the same as a "Riesz space"/"vector lattice."