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In this paper we combine ideas from tolerance orders with recent work on OCinterval orders. We consider representations of posets by unit intervals $I_v$in which the interval endpoints ($L(v)$ and $R(v)$) may be open or closed aswell as the center point ($c(v)$). This yields four types of intervals: $A$(endpoints and center points closed), $B$ (endpoints and center points open),$C$ (endpoints closed, center points open), and $D$ (endpoints open, centerpoints closed). For any non-empty subset $S$ of $\{A,B,C,D\}$, we define an$S$-order as a poset $P$ that has a representation as follows: each element $v$of $P$ is assigned a unit interval $I_v$ of type belonging to $S$, and $x \precy$ if and only if either (i) $R(x) < c(y)$ or (ii) $R(x) = c(y)$ and at leastone of $R(x), c(y)$ is open and at least one of $L(y), c(x)$ is open. Wecharacterize several of the classes of $S$-orders and provide separatingexamples between unequal classes. In addition, for each $S \subseteq\{A,B,C,D\}$ we present a polynomial-time algorithm that recognizes $S$-orders,providing a representation when one exists and otherwise providing acertificate showing it is not an $S$-order.

Tolerance Orders of Open and Closed Intervals