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Monoids of powers in varieties of f-idempotent groupoids
(Hikari, Ltd., 2013)
Celakoska-Jordanova, Vesna
The monoids of powers in varieties of f-idempotent groupoids and commutative f-idempotent groupoids, where f is an irreducible groupoid power with a length at least 3, are constructed. It is shown that these monoids are free over a countable set of irreducible groupoid powers.
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On quaternary associative operations
(Faculty of Technical Sciences Čačak, University of Kragujevac, 2007)
Celakoska-Jordanova, Vesna
Some properties of quaternary associative operations that have neutral sequences are investigated. A characterization of such operations by means of binary primitive operations is established.
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Free power-commutative groupoids
(De Gruyter, 2015-03)
Celakoska-Jordanova, Vesna
A groupoid is called power-commutative if every mono-generated subgroupoid is commutative. The class Pc of power-commutative groupoids is a variety. A description of free objects in this variety and their characterization by means of injective groupoids in Pc are obtained.
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Free power-associative n-ary groupoids
(De Gryter, 2019-01)
Celakoska-Jordanova, Vesna
;
A power-associative n-ary groupoid is an n-ary groupoid G such that for every element a in G, the n-ary subgroupoid of G generated by a is an n-ary subsemigroup of G. The class Pa of power-associative n-ary groupoids is a variety. A description of free objects in this variety and their characterization by means of injective n-ary groupoids in Pa are obtained.
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Free objects in the variety of groupoids defined by the identity xf(x) = f(f(x))
(Hikari, Ltd., 2012)
Celakoska-Jordanova, Vesna
Let V_f denote the variety of groupoids defined by the identity xf(x) = f(f(x)), where f is a fixed nontrivial groupoid power. A description of free objects in this variety and their characterization by means of injective groupoids in V_f are obtained.
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Free groupoids with x^{2}x^{2}=x^{3}x^{3}
(Faculty of Natural Sciences and Mathematics, Skopje, 2004)
Celakoska-Jordanova, Vesna
A description of free objects in the variety V of groupoids defined by the identity x^{2}x^{2}=x^{3}x^{3} is obtained. The following method is used: one of the sides of the identity is considered as "suitable" and the other as "unsuitable" one. First, the left-hand side x^{2}x^{2}is chosen as "suitable" and the set of elements of F (F being an absolutely free groupoid with a basis B) containing no parts that have the form x^{3}x^{3} is taken as a "candidate" for the carrier of the desired free object in V. Continuing this procedure, a V-free object is obtained. Another construction of V-free object is obtained by choosing the right-hand side x^{3}x^{3} as "suitable" one.
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Free groupoids in the class of power left and right idempotent groupoids
(Hikari, Ltd., 2008)
Celakoska-Jordanova, Vesna
The subject of this paper is the class of groupoids such that any groupoid G = (G, ·) of this class has the property: every subgroupoid of G generated by any element a ∈ G satisfies the identity (xx)(yy) = xy. It is shown that this class is a variety. A construction and a characterization of free groupoids in this variety are obtained. The word problem is solvable for this variety.
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Canonical Objects in Classes of (n, V)-Groupoids
(Bulgarian Academy of Sciences - National Committee for Mathematics, 2010)
Celakoska-Jordanova, Vesna
Free algebras are very important in studying classes of algebras, especially varieties of algebras. Any algebra that belongs to a given variety of algebras can be characterized as a homomorphic image of a free algebra of that variety. Describing free algebras is an important task that can be quite complicated, since there is no general method to resolve this problem. The aim of this work is to investigate classes of groupoids, i.e. algebras with one binary operation, that satisfy certain identities or other conditions, and look for free objects in such classes.
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Canonical biassociative groupoids
(Mathematical Institute of the Serbian Academy of Sciences and Arts, 2007)
Janeva, Biljana
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Ilic', Snezhana
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Celakoska-Jordanova, Vesna
In the paper "Free biassociative groupoids", the variety of biassociative groupoids (i.e., groupoids satisfying the condition: every subgroupoid generated by at most two elements is a subsemigroup) is considered and free objects are constructed using a chain of partial biassociative groupoids that satisfy certain properties. The obtained free objects in this variety are not canonical. By a canonical groupoid in a variety V of groupoids we mean a free groupoid (R, ∗) in V with a free basis B such that the carrier R is a subset of the absolutely free groupoid (T_B, ·) with the free basis B and (tu ∈ R ⇒ t, u ∈ R & t∗u = tu). In the present paper, a canonical description of free objects in the variety of biassociative groupoids is obtained.
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Canonical bilaterally commutative groupoids
(Alexandru Ioan Cuza University of Iaşi, 2016)
Celakoska-Jordanova, Vesna
A groupoid G is called bilaterally commutative if it satisfies the identities (xy)z = (yx)z and x(yz) = x(zy). A description of free objects in the variety Bc of bilaterally commutative groupoids and their characterization by means of injective objects in Bc is presented.

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