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The strong endomorphism kernel property for modular p-algebras and for distributive lattices. (English) Zbl 1348.06008

Let \(A\) denote an algebra and let \(\theta\in \mathrm{Con}(A).\) We say that a mapping \(f:A\rightarrow A\) is compatible with \(\theta\) if \((a,b)\in \theta\) implies \((f(a),f(b))\in \theta.\) An endomorphism of \(A\) is called strong if it is compatible with every \(\theta\in \mathrm{Con}(A).\) Now, an algebra \(A\) has the (strong) endomorphism kernel property (shortly EKP or SEKP) if every \(\theta\in \mathrm{Con}(A)\) with \(\theta\neq \nabla_A\) \((=A\times A)\) is the kernel of a (strong) endomorphism of \(A.\) For more details see [T. S. Blyth et al., Commun. Algebra 32, No. 6, 2225–2242 (2004; Zbl 1060.06018); T. S. Blyth and H. J. Silva, Commun. Algebra 36, No. 5, 1682–1694 (2008; Zbl 1148.06005)].
An algebra \(L=(L;\wedge,\vee,^*,0,1)\) of type \((2,2,1,0,0)\) is called a (modular) p-algebra, if \((L;\wedge,\vee,0.1)\) is a (modular) bounded lattice and \(a\mapsto a^*\) means the operation of pseudocomplementation of \(a,\) i.e. \(x\leq a^*\) iff \(x\wedge a=0.\) It is known that the set \(S(L)=\{x\in L: x=x^{**}\}\) of all closed elements of \(L\) forms a Boolean algebra \((S(L);\wedge,+,^*,0,1),\) where \(x+y=(x^*\wedge y^*)^*\) for any \(x, y\in S(L).\) Another significant subset of \(L\) is the dense set \(D(L)=\{x\in L: x^*=0\}.\) Clearly, \(D(L)\) is a dual ideal (= filter) of \(L.\) The third important part of \(L\) is the mapping \(\varphi(L): S(L)\rightarrow F(D(L)),\) where \(F(D(L))\) is the lattice of filters of \(D(L)\) and is defined by \(\varphi(L): a\mapsto D(L)\cap [a^*)\) for every \(a\in S(L).\) If \(L\) is a modular p-algebra, then the associated triple \((S(L), D(L), \varphi(L))\) uniquely determines the algebra \(L\) (see [T. Katrinak and P. Mederly, Algebra Univers. 4, 301–315 (1974; Zbl 0316.06005)]). This result enables to reduce a problem concerning modular p-algebras to two problems: one for Boolean algebras and one for modular lattices with 1.
Main results: The authors characterize the strong congruence kernel property for modular p-algebras applying the method of triple decomposition, and for distributive lattices using Priestley duality. More precisely, (1) Let \(L\) be a non-trivial modular p-algebra. Then \(L\) has SEKP iff (i) \(S(L)\) is a two-element Boolean algebra and (ii) \(D(L)\) has SEKP as a modular lattice with 1. (2) Let \(L\) be a distributive lattice. Then \(L\) has SEKP iff \(L\) is locally finite (i.e. every interval is finite) and there exists \(c\in L\) such that for every \(x<c\) or \(c>x\) the intervals \([x,c]\) and \([c,x]\) (if \(x>c\)) are finite Boolean lattices.

MSC:

06D15 Pseudocomplemented lattices
08A30 Subalgebras, congruence relations
08A35 Automorphisms and endomorphisms of algebraic structures
06D10 Complete distributivity
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