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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)
##### Keywords:
posets; groupoids; pogroupoid; pg-algebra; Harris diagrams
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