## 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

### Citations:

Zbl 1060.06018; Zbl 1148.06005; Zbl 0316.06005
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### References:

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