### Idempotent Subreducts of Semimodules over Commutative Semirings

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We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity xⁿ ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found.

A groupoid $H$ is a homomorphic image of a subdirectly irreducible groupoid $G$ (over its monolith) if and only if $H$ has a smallest ideal.

We present an elementary proof (purely in equational logic) that distributive groupoids are symmetric-by-medial.

We study a class of strongly solvable modes, called differential modes. We characterize abelian algebras in this class and prove that all of them are quasi-affine, i.e., they are subreducts of modules over commutative rings.

We prove that, for any prime $p$, there are precisely $2{p}^{4}-{p}^{3}-{p}^{2}-3p-1$ medial quasigroups of order ${p}^{2}$, up to isomorphism.

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct.

We study groupoids satisfying the identities x·xy = y and x·yz = xy·xz. Particularly, we focus our attention at subdirectlyirreducible ones, find a description and charecterize small ones.

We prove that a finite unary algebra with at least two operation symbols is a homomorphic image of a (finite) subdirectly irreducible algebra if and only if the intersection of all its subalgebras which have at least two elements is nonempty.

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