The structure of completely meet irreducible congruences in strongly Fregean algebras

A special class of congruences on $κ$-frames

Madden has shown that in contrast to the situation with frames, the smallest dense quotient of a $κ$-frame need not be Boolean. We characterise these so-called d-reduced $κ$-frames as those which may be embedded as a generating sub-$κ$-frame of a Boolean frame. We introduce the notion of the closure of a $κ$-frame congruence and call a congruence clear if it is the largest congruence with a given closure. These ideas are used to prove $κ$-frame analogues of known results concerning Boolean frame quotients. In particular, we show that d-reduced $κ$-frames are precisely the quotients of $κ$-fram…

具有大于3/32的同余密度的格

Lattices with congruence densities larger than $3/32$

研究了同余密度大于3/32的有限格的性质，证明此类格的join-irreducible和meet-irreducible元素数量相同，并给出了相关同余数的推论。

Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties

We construct a diagram D, indexed by a finite partially ordered set, of finite Boolean semilattices and (v,0,1)-embeddings, with top semilattice $2^4$, such that for any variety V of algebras, if D has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra $U$ in V such that the congruence lattice of $U$ contains, as a 0,1-sublattice, the five-element modular nondistributive lattice $M_3$. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive. Using earlier work of K.A. Kearnes an…

Congruences on completely inverse $AG^{**}$-groupoids

By a completely inverse $AG^{**}$-groupoid we mean an inverse $AG^{**}$-groupoid $A$ satisfying the identity $xx^{-1}=x^{-1}x$, where $x^{-1}$ denotes a unique element of $A$ such that $x=(xx^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}.$ We show that the set of all idempotents of such groupoid forms a semilattice and the Green's relations $\mathcal{H,L, R,D}$ and $\mathcal{J}$ coincide on $A$. The main result of this note says that any completely inverse $AG^{**}$-groupoid meets the famous Lallement's Lemma for regular semigroups. Finally, we show that the Green's relation $\mathcal{H}$ is both the le…

Supernilpotent Taylor algebras are nilpotent

We develop the theory of the higher commutator for Taylor varieties. A new higher commutator operation called the hypercommutator is defined using a type of invariant relation called a higher dimensional congruence. The hypercommutator is shown to be symmetric and satisfy an inequality relating nested terms. For a Taylor algebra the term condition higher commutator and the hypercommutator are equal when evaluated at a constant tuple, and it follows that every supernilpotent Taylor algebra is nilpotent. We end with a characterization of congruence meet-semidistributive varieties in terms of the…

Weyl群的格结构：通过预投射代数的表示论

Lattice structure of Weyl groups via representation theory of preprojective algebras

本文利用预投射代数Π的表示论，研究有限Weyl群W上弱序的格同余的组合结构，建立了join-不可约同余、join-不可约（或meet-不可约）元素、不可分解τ-刚性（或τ⁻-刚性）模与Π的层之间的自然双射，并证明了Hasse箭图的两种标号方式一致。