### A certain Galois connection and weak automorphisms

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An algorithm is given to decompose an automorphism of a finite vector space over ℤ₂ into a product of transvections. The procedure uses partitions of the indexing set of a redundant base. With respect to tents, i.e. finite ℤ₂-representations generated by a redundant base, this is a decomposition into base changes.

Let $A:=(A,\to ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations ${\alpha}_{p}:x\mapsto (p\to x)$, which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of $A$. This semilattice is isomorphic to the semilattice of finitely generated filters of $A$, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of $A$. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of $A$, and the...

Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.

By an equivalence system is meant a couple $\mathcal{A}=(A,\theta )$ where $A$ is a non-void set and $\theta $ is an equivalence on $A$. A mapping $h$ of an equivalence system $\mathcal{A}$ into $\mathcal{B}$ is called a class preserving mapping if $h\left({\left[a\right]}_{\theta}\right)={\left[h\left(a\right)\right]}_{\theta {}^{\text{'}}}$ for each $a\in A$. We will characterize class preserving mappings by means of permutability of $\theta $ with the equivalence ${\Phi}_{h}$ induced by $h$.

In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a “well-behaved core”, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this “core” is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term $t\left(x\right)$) in every member of the given variety. Here, we try to give a unified account of this phenomenon....

An M-Set is a unary algebra $\langle X,M\rangle $ whose set $M$ of operations is a monoid of transformations of $X$; $\langle X,M\rangle $ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $\langle X,M\rangle $ if the congruence lattice of $\langle X,M\rangle $ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\Pi \left(L\right)$ is introduced here. $\Pi \left(L\right)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\Pi \left(L\right)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi $-product lattice. A $\Pi $-product...

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.

Some decompositions of general incidence structures with regard to distinguished components (modular or simple) are considered and several structure theorems for them are deduced.