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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.

Definability in the lattice of equational theories of semigroups.

J. Jezek, R. McKenzie (1993)

Semigroup forum

Definability of principal congruences in equivalential algebras

PaweŁ Idziak, Andrzej Wroński (1998)

Colloquium Mathematicae

Demi-groupes, espaces affines et categories gauches

André Batbedat (1977)

Fundamenta Mathematicae

Demi-Semi-Primal Algebras and Mal'cev- Type Conditions.

Robert W. Quackenbusch (1971)

Mathematische Zeitschrift

Der Defkt endlicher abelscher Gruppen.

H.K. Kaiser (1974)

Monatshefte für Mathematik

Diamond identities for relative congruences

Gábor Czédli (1995)

Archivum Mathematicum

For a class $K$ of structures and $A \in K$ let ${C o n}^{*} (A)$ resp. ${C o n}^{K} (A)$ denote the lattices of $*$ -congruences resp. $K$ -congruences of $A$ , cf. Weaver [25]. Let ${C o n}^{*} (K) : = I {{C o n}^{*} (A) : A \in K}$ where $I$ is the operator of forming isomorphic copies, and ${C o n}^{r} (K) : = I {{C o n}^{K} (A) : A \in K}$ . For an ordered algebra $A$ the lattice of order congruences of $A$ is denoted by ${C o n}^{<} (A)$ , and let ${C o n}^{<} (K) : = I {{C o n}^{<} (A) : A \in K}$ if $K$ is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by $Q^{s}$ and $P$ , respectively. Let $λ$ be a lattice identity and let $Σ$ be a set of lattice identities. Let $Σ ⊧_{c} λ (r; Q^{s}, P)$ denote...

Diassociativity is not finitely based relative to power associativity

Tomasz Kowalski (2010)

Commentationes Mathematicae Universitatis Carolinae

We show that the variety of diassociative loops is not finitely based even relative to power associative loops with inverse property.

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