格的有界同态与格生成的纤维积
Bounded homomorphisms and finitely generated fiber products of lattices
摘要：
我们研究了格的纤维积何时生成，并得到了格上有界格同态满足Dean条件（D）的一个新的刻画.特别地，所有的非线性表示格和满足惠特曼条件的格都满足（D）。对于格满同态$g\colon A\to D$，$h\colon B\to D$，其中$A$，$B$是逆生成的且$D$满足（D），我们证明了：如果$g$和$h$是有界的，则它们的纤维积（拉回）$C=\{(a,b)\in A\times B\ |\ g(a)=h(b)\}$是逆生成的.虽然一般情况下反之不成立，但当$A$和$B$空闲时，它确实成立。作为结果，我们得到了一个（指数时间）算法，以确定的有界性的双表示格和他们的双生成子格满足（D）。这推广了Freese和Nation的一个未发表的结果。
英文摘要：
We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms $g\colon A\to D$, $h\colon B\to D$, where $A$, $B$ are finitely generated and $D$ satisfies (D), we show the following: If $g$ and $h$ are bounded, then their fiber product (pullback) $C=\{(a,b)\in A\times B\ |\ g(a)=h(b)\}$ is finitely generated. While the converse is not true in general, it does hold when $A$ and $B$ are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.
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