Series-parallel poset

Series-parallel poset --- параллельно-последовательное чу-множество.

Let [math]\displaystyle{ P_{1} = (V_{1}, \lt _{1}) }[/math] and [math]\displaystyle{ P_{2} = (V_{2}, \lt _{2}) }[/math] be finite posets with [math]\displaystyle{ V_{1} \cap V_{2} = \emptyset }[/math]. The parallel composition [math]\displaystyle{ P_{1} + P_{2} = (V, \lt _{+}) }[/math] of [math]\displaystyle{ P_{1}, P_{2} }[/math] is defined by [math]\displaystyle{ V = V_{1} \cup V_{2} }[/math], and [math]\displaystyle{ u \lt v }[/math] iff ([math]\displaystyle{ u,v \in V_{1} }[/math] and [math]\displaystyle{ u \lt _{1} v }[/math]) or ([math]\displaystyle{ u, v \in V_{2} }[/math] and [math]\displaystyle{ u \lt _{2} v }[/math]). The series composition [math]\displaystyle{ P_{1} \ast P_{2} = (V, \lt _{\ast}) }[/math] of [math]\displaystyle{ P_{1}, P_{2} }[/math] is defined by [math]\displaystyle{ V = V_{1} \cup V_{2} }[/math], and [math]\displaystyle{ u \lt v }[/math] iff ([math]\displaystyle{ u,v \in V_{1} }[/math] and [math]\displaystyle{ u \lt _{1} v }[/math]) or ([math]\displaystyle{ u, v \in V_{2} }[/math] and [math]\displaystyle{ u \lt _{2} v }[/math]) or [math]\displaystyle{ u \in V_{1} }[/math] and [math]\displaystyle{ v \in V_{2} }[/math]. [math]\displaystyle{ P = (V, \lt ) }[/math] is series-parallel poset if [math]\displaystyle{ |V| = 1 }[/math] or [math]\displaystyle{ P }[/math] is obtained by [math]\displaystyle{ P_{1} + P_{2} }[/math] or [math]\displaystyle{ P_{1} \ast P_{2} }[/math] of smaller series-parallel posets [math]\displaystyle{ P_{1} }[/math], [math]\displaystyle{ P_{2} }[/math], i.e. the class of series-parallel posets is the smallest class which contains the one-element poset and is closed under [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \ast }[/math].