$$S$$-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

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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