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Algebras associated with posets. (English) Zbl 0985.06001
The paper studies algebraic structures which are associated with posets. A groupoid \(S(\cdot)\) is called a pogroupoid, if \(x\cdot y\in \{x,y\}\), \(x\cdot(y\cdot z)= y\cdot x\) and \((x\cdot y)\cdot (y\cdot z)= (x\cdot y)\cdot z\) for any three elements \(x\), \(y\), \(z\) of \(S\). A pogroupoid \(S(\cdot)\) is associated with a poset \(S(\leq)\) in such a way that \(x\cdot y= x\) for \(y\leq x\) and \(x\cdot y= y\) otherwise. If a pogroupoid \(S(\cdot)\) and a field \(K\) are given, then a pg-algebra \(KS\) over the field \(K\) is defined. These algebras are studied. In particular, the authors study the interrelations between pg-algebras and Harris diagrams of posets and further quotient pg-algebras. The Harris diagram of a poset \(S(\leq)\) is a graph whose vertex set is \(S\) and in which two vertices are adjacent if and only if they are incomparable in \(S(\leq)\).

MSC:
06A06 Partial orders, general
20N02 Sets with a single binary operation (groupoids)
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