Barycentric coordinates provide solutions to the problem of expressing an element a compact convex set as combination finite number extreme points set. They have been studied widely within geometric literature, typically in response demands interpolation, numerical analysis and computer graphics. In this note we bring algebraic perspective problem, based on barycentric algebras. We focus discussion relations between different subclasses partitions unity, one arising context coordinates,...

Abstract We present a construction of all finite indecomposable involutive solutions the Yang–Baxter equation multipermutational level at most 2 with abelian permutation group. As consequence, we obtain formula for number such fixed elements. also describe some properties automorphism groups in this case; particular, show they are regular groups.

We study indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with cyclic permutation groups (cocyclic solutions).In particular, we show that there is no one-to-one correspondence between cocyclic and braces which contradicts recent results in [19].

We prove that the class K(σ) of all algebraic structures signature σ is Q-universal if and only there a K ⊆ such problem whether finite lattice embeds into K-quasivarieties undecidable.

This paper is devoted to the semilattice ordered $\mathcal{V}$ -algebras of form (A, Ω, + ), where a join-semilattice operation and Ω) an algebra from some given variety . We characterize free algebras using concept extended power algebras. Next we apply result describe lattice subvarieties in relation

We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes such groups, as semi-commutativity, we construct a

We find a syntactic characterization of the class lattices embeddable into convexity posets which are trees. The implies that this forms finitely based variety.

Abstract In a natural way we can “lift” any operation defined on set A to an the of all non-empty subsets and obtain from algebra (A, Ω) its power subsets. this paper investigate extended algebras (power with one additional semilattice operation) modes (entropic idempotent algebras). We describe some congruence relations these such that their quotients are idempotent. Such congruences determine class non-trivial subvarieties variety ordered (modals).

We investigate a class of non-involutive solutions the Yang–Baxter equation which generalize derived (self-distributive) solutions. In particular, we study generalized multipermutation in this class. show that (permutation) groups such are nilpotent. formulate results language biracks allows us to apply universal algebra tools.

In a natural way, we can "lift" any operation defined on set A to an the of all non-empty subsets and obtain from algebra ( $${A, \Omega}$$ ) its power subsets. G. Grätzer H. Lakser proved that for variety $${\mathcal{V}}$$ , $${\mathcal{V}\Sigma}$$ generated by algebras in satisfies precisely consequences linear identities true . For certain types algebras, sets their subalgebras form algebras. They are called subalgebras. this paper, partially solve long-standing problem concerning...

Abstract The main aim of this paper is to describe the free objects in arbitrary varieties modals (semilattice ordered idempotent and entropic algebras) give some new representations modals.

This paper gives the construction of free medial quandles as well n-symmetric and m-reductive quandles.

We describe all subdirectly irreducible medial quandles. show that they fall within one of four disjoint classes. In particular, in the finite case are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation non-connected

We study connections between closure operators on an algebra (A, Ω) and congruences the extended power defined same algebra. use these to give alternative description of lattice all subvarieties semilattice ordered algebras.