Gábor Czédli
For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.
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We attach to each $\langle 0, \vee \rangle$-semilattice a graph $\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elements of $\boldsymbol{S}$ and whose edges correspond to the reflexive dependency relation. We study properties of the graph $\boldsymbol{G}_{\boldsymbol{S}}$ both when $\boldsymbol{S}$ is a join-semilattice and when it is a lattice. We call a $\langle 0, \vee \rangle$-semilattice $\boldsymbol{S}$ particle provided that the set of its join-irreducible elements join-generates $\boldsymbol{S}$ and it satisfies DCC. We prove that the congruence lattice of a parti…
Semilattices with a congruence
本文定义了特殊化半格，证明其范畴与带有一个同余的半格范畴同构且等价于半格满同态范畴，并借此回顾了一般关系系统间满射同态的内部刻画。
Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties
我们构造了一个有限布尔半格和（v，0，1）-嵌入的图D，它由一个有限偏序集作索引，且具有顶半格$2^4$，使得对于任意代数簇V，如果D关于同余格函子有一个提升，通过V中的代数和同态，则在V中存在一个代数$U$，使得$U$的同余格包含，五元模非分配格$M_3$作为0，1-子格。特别地，V有一个代数，它的同余格既不是并半分配的，也不是交半分配的。使用K.A.的早期工作。Kearnes和A.Szendrei的结果，我们还推出了V没有非平凡的同余格恒等式.特别地，不存在从有限布尔半格和（v，0，1）-嵌入到格和格嵌入的函子F，使得组合Con F等价于单位元（其中Con表示同余格函子），从而否定地解决了P. Pudlak在1985年提出的关于同余格问题的函子解的存在性问题.
The pairwise distributive law of semilattice congruences
本文证明半格同余格相对于主同余满足分配形式，建立了任意同余族（含无限集）的“成对分配律”。
Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part I
我们表明，对于每一个准品种K的结构（其中功能和关系是允许的）有一个半格S与运营商，使格的准方程理论的K（对偶格的子准品种的K）是同构的Con（S，+，0，F）。结果，发现了拟方程理论格上自然拟内部算子的新的限制。
Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part II
第一部分证明了对每一个结构的拟簇K（它可能既有运算又有关系），存在一个具有算子的半格S，使得K的拟方程理论格（K的子拟簇格的对偶）同构于Con（S，+，0，F）。已知如果S是一个有0（且无算子）的并半格，则存在一个拟簇Q使得Q的理论格同构于Con（S，+，0）.证明了：若S是一个既有0又有1的半格，且G上有一个作用于S的算子群G，且G中的每个算子都固定0和1，则存在一个拟簇W，使得W的拟方程理论格同构于Con（S，+，0，G）.
Congruences on completely inverse $AG^{**}$-groupoids
完全逆$AG^{**}$ -广群是指满足恒等式$xx^{-1}=x^{-1}x$的逆$AG^{**}$ -广群$A$，其中$x^{-1}$表示$A$的唯一元素，使得$x=(xx^{-1})x$和$x^{-1}=(x^{-1}x)x^{-1}.$我们证明了这样的广群的所有幂等元的集合形成半格，并且格林关系$\mathcal{H,L, R,D}$和$\mathcal{J}$在$A$上重合.本文的主要结果是：任何完全逆的$AG^{**}$ -广群满足正则半群的著名的Lallement引理。最后证明了格林关系$\mathcal{H}$是任意完全逆$AG^{**}$ -广群上的最小半格同余和最大幂等元分离同余.
Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question, whether $S$ can be infinite at all.
Critical points of pairs of varieties of algebras
对于代数类V，用Conc（V）表示同构于A的所有紧同余的半格Conc（A）的所有半格的类。对于代数类V1和V2，我们用crit（V1，V2）表示Conc（V1）中的一个半格的最小基数，如果它存在，它不在Conc（V2）中，否则为无穷大。本文证明了一个具有范畴性质的一般定理，它意味着对于所有的非线性生成的同余分配簇V1和V2，crit（V1，V2）要么是有限的，要么是对某个自然数n的aleph_n，要么是无穷大的.我们还找到了两个可生成的模格簇V1和V2，使得crit（V1，V2）=aleph_1，从而回答了J.Tuma和F.威龙
The possible values of critical points between strongly congruence-proper varieties of algebras
我们用Conc（A）表示一个（泛）代数A的所有Conc生成的同余的半格，并且我们定义Conc（V）为所有Conc（A）的所有同构副本的类，对于A在V中，对于任何代数簇V。设V和W是代数的局部有限簇，使得对于V中的每个有限代数A，在同构之前，W中只有2/3个B，使得A和B有同构的同余格，并且每个这样的B是有限的.如果Conc（V）不包含在Conc（W）中，则存在基数为aleph 2的半格在Conc（V）-Conc（W）中.我们的结果推广到一阶结构的拟簇，具有许多关系符号和相对同余格。特别地，如果W是代数的一个非平凡生成簇，则这发生在W省略了驯服的同余理论类型1和类型5的情况下;反过来，这发生在W满足非平凡的同余恒等式的情况下。绑定aleph 2是尖锐的。
A solution to Dilworth's Congruence Lattice Problem
我们构造了一个分配代数格D，它不同构于任何格的同余格。这解决了一个长期存在的开放问题，传统上归因于R。P. Dilworth，40年代的。格点D有紧顶元和aleph Ω +1紧元。我们的结果推广到所有具有最大元素的并半格的多项式可定义结构的代数。
Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly $2^{n-5}$ congruences.
Let $S$ be a semigroup and $\mathbb F$ be a field. For an ideal $J$ of the semigroup algebra ${\mathbb F}[S]$ of $S$ over $\mathbb F$, let $\varrho _J$ denote the restriction (to $S$) of the congruence on ${\mathbb F}[S]$ defined by the ideal $J$. A semigroup $S$ is called a permutable semigroup if $α\circ β=β\circ α$ is satisfied for all congruences $α$ and $β$ of $S$. In this paper we show that if $S$ is a semilattice or a rectangular band then $φ_{\{S;{\mathbb F}\}}:\ J\mapsto \varrho _J$ is a homomorphism of the semigroup $(Con ({\mathbb F}[S]);\circ )$ into the relations semigroup $({\cal…
A specialization semilattice is a join semilattice together with a coarser preorder $ \sqsubseteq $ satisfying an appropriate compatibility condition. If $X$ is a topological space, then $(\mathcal P(X), \cup, \sqsubseteq )$ is a specialization semilattice, where $ x \sqsubseteq y$ if $x \subseteq Ky$, for $x,y \subseteq X$, and $K$ is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. For short, the notion is useful since it allows us to consider a relation of "being generated by" with no need to requ…
We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets; consequently, the same holds for antiortholattices. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice, pseudo--Kleene algebra or antiortholattice can have any number of congruences between $2$ and its number of subsets, regardless of its number of ideals.
By a twenty year old result of Ralph Freese, an $n$-element lattice $L$ has at most $2^{n-1}$ congruences. We prove that if $L$ has less than $2^{n-1}$ congruences, then it has at most $2^{n-2}$ congruences. Also, we describe the $n$-element lattices with exactly $2^{n-2}$ congruences.
Let $λ$ and $κ$ be cardinal numbers such that $κ$ is infinite and either $2\leq λ\leq κ$, or $λ=2^κ$. We prove that there exists a lattice $L$ with exactly $λ$ many congruences, $2^κ$ many ideals, but only $κ$ many filters. Furthermore, if $λ\geq 2$ is an integer of the form $2^m\cdot 3^n$, then we can choose $L$ to be a modular lattice generating one of the minimal modular nondistributive congruence varieties described by Ralph Freese in 1976, and this $L$ is even relatively complemented for $λ=2$. Related to some earlier results of George Grätzer and the first author, we also prove that if $…
On Congruences on Ultraproducts of Algebraic Structures
本文研究代数结构超积上的同余，构造了各结构同余格的超积到超积结构同余格的嵌入，并给出相应商代数同构及对角限制公式。
We characterize those nilpotent algebras of prime power order and finite type in congruence modular varieties that have infinitely many polynomially inequivalent congruence preserving expansions.
From objects to diagrams for ranges of functors
设A，B，S是范畴，F：A-->S和G：B-->S是函子.我们假设对于A中的“许多”对象a，存在B中的对象b，使得F（a）同构于G（b）。我们建立了一个一般的框架，在这个框架下，有可能把这个陈述转化为A的图。这些图都是以偏序集为索引的，其中每个主理想都是并半格，且任何有限子集的所有上界的集合都是一个可生成的上子集。以下是一些推论，特别是：（1）格雷泽-施密特定理（Grätzer-Schmidt Theorem），它指出每个代数格同构于某个代数的同余格，可以推广到代数格的有限偏序集索引图和保紧的完全并同态（如果有足够大的基数，则没有有限性限制）。(2)在许多情况下，两个局部有限拟簇之间的相对临界点要么小于aleph Ω，要么等于无穷大。(3)基数为aleph 1的格可能没有任何可置换的、保同余的扩张。
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