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Covering graphs and subdirect decompositions of partially ordered sets. (English) Zbl 0603.06001
The covering graph C($${\mathcal L})$$ of a partially ordered set (poset) $${\mathcal L}=(L,\leq)$$ is defined to be the undirected graph whose edges are those pairs (a,b) of L for which either a covers b or vice versa and whose vertices are the elements of L. All posets $${\mathcal L}$$ dealt with in the present paper are assumed to be almost discrete, i.e. whenever a,b$$\in L$$, $$a<b$$, then there are elements $$a_ 0,a_ 1,...,a_ n\in L$$ such that $$a_ 0=a$$, $$a_ n=b$$ and $$a_ i$$ covers $$a_{i-1}$$, $$i=1,2,...,n$$. Let $$\{$$ $${\mathcal L}_ i\}_{i\in I}$$ be a system of posets. By $$\prod_{i\in I}{\mathcal L}_ i={\mathcal L}$$ or (sub)$$\prod_{i\in I}{\mathcal L}_ i$$ there is denoted the direct or subdirect product of $${\mathcal L}_ i$$, $$i\in I$$, respectively. An isomorphism $$\phi$$ of a poset $${\mathcal L}'$$ into $${\mathcal L}$$ such that $$\phi$$ ($${\mathcal L}')=(sub)\prod_{i\in I}{\mathcal L}_ i$$ is said to be a subdirect product representation of $${\mathcal L}'$$. The subdirect product representation $$\phi$$ of $${\mathcal L}'$$ is said to induce a subdirect product representation of the graph C($${\mathcal L}')$$ if $$\phi$$ : C($${\mathcal L}')\to \prod_{i\in I}C({\mathcal L}_ i)$$ is a subdirect product representation of the graph C($${\mathcal L}')$$. Analogous notions and notations are defined for graphs. Let us have a subdirect decomposition $$\phi$$ : C($${\mathcal L})\to (sub)\prod_{i\in I}{\mathcal G}_ i$$ of the covering graph C($${\mathcal L})$$. Then the condition that ($$\alpha)\phi$$ induces a subdirect decomposition of $${\mathcal L}$$ need not be valid in general. The main result of the present paper is the statement that the following condition ($$\beta)$$ is necessary for ($$\alpha)$$ to be valid: ($$\beta)$$ If K is a saturated subset of L such that $${\mathcal K}=(K,\leq)$$ is isomorphic to $${\mathcal L}_ 1$$, then there is $$i\in I$$ such that $$\phi_ j(K)=1$$ for each $$j\in I\setminus \{i\}$$. ($${\mathcal L}_ 1$$ is a poset with four elements $$a_ 1,a_ 2,b_ 1,b_ 2$$ such that $$a_ i$$ is covered by $$b_ j$$ $$(i,j=1,2)$$ and there are no other covering relations in $${\mathcal L}_ 1.)$$ Another setting of this theorem is the following: Let $$\psi$$ be a subdirect product representation of the graph C($${\mathcal L}')$$. If $$\psi$$ induces a subdirect product representation of $${\mathcal L}'$$, then the condition ($$\beta)$$ is fulfilled. If, especially, $${\mathcal L}'$$ is connected and $$\psi$$ is a weak direct product (or direct product) representation of C($${\mathcal L}')$$, then $$\psi$$ induces a weak direct product (or direct product) representation of $${\mathcal L}'$$ iff the condition ($$\beta)$$ holds. If $${\mathcal L}$$ is a semilattice and $$\psi$$ as in the preceding proposition, then the condition ($$\beta)$$ is valid. Some counterexamples complete the paper.
Reviewer: F.Šik

##### MSC:
 06A06 Partial orders, general 05C75 Structural characterization of families of graphs
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##### References:
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