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\(S\)-pasted sum of lattices. (English. Russian original) Zbl 0746.06002

Algebra Logic 29, No. 4, 301-309 (1990); translation from Algebra Logika 29, No. 4, 452-463 (1990).
Dropping the finite length property, the author generalizes the notion of \(S\)-pasted system (sum) of lattices, introduced by C. Herrmann [Math. Z. 130, 255-274 (1973; Zbl 0275.06007)]. Using this new construction, the author characterizes the congruence lattices of connected monounary algebras with a loop. In more detail, a lattice \(L\) is said to be a (generalized) \(S\)-pasted sum of lattices \(L_ s\), \(s\in S\), if (i) \(S\) is a lattice, (ii) \(L=\bigcup(L_ s:s\in S)\), (iii) \(L_ s\) is a convex sublattice of \(L\) for every \(s\in S\), (iv) if \(x,y\in L\) and \(x\in L_ s\), \(y\in L_ t\), then \(x\lor y\in L_{s\lor t}\) and \(x\land y\in L_{s\land t}\), (v) \(s\neq t\) implies \(L_ s\cap L_ t\not\in \{L_ s,L_ t\}\). A monounary algebra \((A;f)\) is called connected if for any \(x,y\in A\) there exist nonnegative integers \(m, n\) such that \(f^ m(x)=f^ n(y)\;(f^ 0(x)=x,\;f^{n+1}(x)=f(f^ n(x)))\). An element \(a\in A\) is said to be a loop of \((A;f)\), if \(f(a)=a\).
Main results: (1) Let \(S\) be a lattice and \(M=\{L_ s:s\in S\}\) a system of lattices. Then there exists a lattice which is a generalized \(S\)-sum of lattices \(L_ s\), \(s\in S\) if and only if five conditions of adhesion are satisfied. (2) Let \((A;f)\) be a connected monounary algebra with a loop. Then \(\hbox{Con}(A)\) is a generalized \(S\)-sum of direct products of equivalence lattices. Moreover, \(S=\{\alpha\in \hbox{Con}(A):(\alpha^*)_ +=\alpha\}\) is a sublattice of \(\hbox{Con}(A)\) and \(\alpha^*=\sup(b:b \hbox{ covers } \alpha)\) or \(\alpha^*=\alpha\) whenever no cover of \(\alpha\) exists. (\(\alpha_ +\) is defined in a dual manner).

MSC:

06B05 Structure theory of lattices
08A30 Subalgebras, congruence relations
08A60 Unary algebras

Citations:

Zbl 0275.06007
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References:

[1] C. Herrmann, ”S-verklebte Summen von Verbänden,” Math. Z.,130, 255–274 (1973). · Zbl 0275.06007
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