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### 2-normalization of lattices

Czechoslovak Mathematical Journal

Let $\tau$ be a type of algebras. A valuation of terms of type $\tau$ is a function $v$ assigning to each term $t$ of type $\tau$ a value $v\left(t\right)\ge 0$. For $k\ge 1$, an identity $s\approx t$ of type $\tau$ is said to be $k$-normal (with respect to valuation $v$) if either $s=t$ or both $s$ and $t$ have value $\ge k$. Taking $k=1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$-normal (with respect to the valuation $v$) if all its identities are $k$-normal. For any variety $V$, there is a least...

Semigroup forum

### A multiplication of $e$-varieties of orthodox semigroups

Archivum Mathematicum

We define semantically a partial multiplication on the lattice of all e–varieties of regular semigroups. In the case that the first factor is an e–variety of orthodox semigroups we describe our multiplication syntactically in terms of biinvariant congruences.

### A multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties

Archivum Mathematicum

A multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties is described both semantically and syntactically. The associativity of the multiplication is also proved.

Semigroup forum

### A note on normal varieties of monounary algebras

Czechoslovak Mathematical Journal

A variety is called normal if no laws of the form $s=t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $\left(A,f\right)$ and let $V$ be a non-trivial non-normal element of $L$. Then $V$ is of the form $\mathrm{M}od\left({f}^{n}\left(x\right)=x\right)$ with some $n>0$. It is shown that the smallest normal variety containing $V$ is contained in $\mathrm{H}SC\left(\mathrm{M}od\left({f}^{mn}\left(x\right)=x\right)\right)$ for every $m>1$ where $\mathrm{C}$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal elements of...

### A note on permutability in varieties

Časopis pro pěstování matematiky

### A reduction theorem for ring varieties whose subvariety lattice is distributive

Discussiones Mathematicae - General Algebra and Applications

We prove a theorem (for arbitrary ring varieties and, in a stronger form, for varieties of associative rings) which basically reduces the problem of a description of varieties with distributive subvariety lattice to the case of algebras over a finite prime field.

Semigroup forum

### A ternary variety generated by lattices

Commentationes Mathematicae Universitatis Carolinae

### Algebras with principal tolerances

Mathematica Slovaca

### All subvarieties of ${ℒ}_{pq}$ have finite bases of identities.

Sibirskij Matematicheskij Zhurnal

Semigroup forum

Semigroup forum

Semigroup forum

### Antiatomic retract varieties of monounary algebras

Czechoslovak Mathematical Journal

### Binary relations on the monoid of V-proper hypersubstitutions

Discussiones Mathematicae - General Algebra and Applications

In this paper we consider different relations on the set P(V) of all proper hypersubstitutions with respect to a given variety V and their properties. Using these relations we introduce the cardinalities of the corresponding quotient sets as degrees and determine the properties of solid varieties having given degrees. Finally, for all varieties of bands we determine their degrees.

### Characterizing pure, cryptic and Clifford inverse semigroups

Czechoslovak Mathematical Journal

An inverse semigroup $S$ is pure if $e={e}^{2}$, $a\in S$, $e implies ${a}^{2}=a$; it is cryptic if Green’s relation $ℋ$ on $S$ is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and...