Commutative Idempotent Research Articles

A category equivalence for odd Sugihara monoids and its applications

An odd Sugihara monoid is a residuated distributive lattice-ordered commutative idempotent monoid with an order-reversing involution that fixes the monoid identity. The main theorem of this paper establishes a category equivalence between odd Sugihara monoids and relative Stone algebras. In combination with known results, it swiftly determines which varieties of odd Sugihara monoids are [strongly] amalgamable and which have the strong [or weak] epimorphism-surjectivity property. In particular, the full variety is shown to have all of these properties. The results extend, with slight modification, to the case where the algebras are bounded. Logical applications include immediate answers to some questions about projective and finite Beth definability and interpolation in the uninorm-based logic IUML, its boundless fragment and all of their extensions.

Idempotent semigroups and tropical algebraic sets

The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup structure. We address the question of the geometry of idempotent semigroups, in particular, tropical algebraic sets carrying the structure of a commutative idempotent semigroup. We show that commutative idempotent semigroups are contractible, that systems of tropical polynomials, formed from univariate monomials, define subsemigroups with respect to coordinate-wise tropical addition (maximum); and, finally, we prove that the subsemigroups in the Euclidean space which are either tropical hypersurfaces, or tropical curves in the plane or in the three-space have the above polynomial description.

Algebraic Structure of TCI-Groupoids: Classes, Clones, Constructions

Some aspects of the algebraic structure of TCI-groupoids (totally commutative idempotent groupoids) are presented. Among other things, a full description of the corresponding binary clones is given. To investigate the algebraic structure of TCI-groupoids and their clones the families of two-generated subgroupoids are used. Next the algebraic structure of these families is studied in detail. We present a special construction which allows us to build each finite TCI-groupoid using the two-element semilattice as a kind of elementary "brick". Furthermore it is proved that the class of all regular TCI-groupoids satisfying binary nonbalanced identities is a nonaxiomatizable proper subclass of the class of all regular TCI-groupoids.

Amenability constants for semilattice algebras

For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ 1(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We also give example of a commutative Clifford semigroups G n whose semigroup algebras ℓ 1(G n ) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9.

On finite X-semilattices of unions

Using a characteristic family of sets, a characteristic mapping, and basis sources of an X-semilattice of unions D, we characterize the class Σ(X, m) consisting of all finite X-semilattices of unions that are isomorphic to a semilattice D given in advance. For a finite set X, the number of elements in the considered class is found. Commutative semigroups of idempotents are known to play a significant role in semigroup theory (see [25, 26]). Moreover, any commutative idempotent semigroup is isomorphic to some X-semilattice of unions (see [26]), whereas X-semilattices play an especially important role in studying many abstract properties of complete semigroups of binary relations (see [1–4, 7–24]).

Minimal varieties of residuated lattices

In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Finally, we study the connections with the subvariety lattice of residuated bounded-lattices. We modify the construction mentioned above to obtain a continuum of idempotent, representable minimal varieties of residuated bounded-lattices and illustrate how the existing construction provides continuum many covers of the variety generated by the three-element non-integral residuated bounded-lattice.

Equational theories of tropical semirings

This paper studies the equational theories of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max,+) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

A Fully Equational Proof of Parikh's Theorem

We show that the validity of Parikh's theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of μ-term equations of continuous commutative idempotent semirings.

Equational Theories of Tropical Semirings

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the <em> (max,+) semiring</em> and the <em>tropical semiring</em>. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

A Fully Equational Proof of Parikh's Theorem

We show that the validity of Parikh's theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of mu-term equations of continuous commutative idempotent semirings.

Numbers implying algebraic structures

We prove in this paper that a commutative idempotent groupoid is a proper Plonka sum of affine spaces over GF(3) if and only if it has 27 essentially 4-ary term functions. This means, in another terminology, that the number 27 is the characteristic number of those sums in the variety of all commutative idempotent groupoids. Using this result, we also show that a medial idempotent groupoid with three essentially binary operations contains a subgroupoid term equivalent to either the five-element affine space over GF(5) or a Plonka sum of a trivial groupoid and the three-element affine space over GF(3).

The minimal extension of sequences III. On problem 16 of Grätzer and Kisielewicz

The main result of this paper is a description of totally commutative idempotent groupoids. In particular, we show that if an idempotent groupoid (G,·) has precisely m ≥ 2 distinct essentially binary polynomials and they are all commutative, then G contai

On commutative idempotent rings

We study the problem when a ring which is an extension of a commutative idempotent ring by a commutative idempotent ring is commutative. In particular, we answer Sands' question showing that the class of commutative idempotent rings whose every homomorphic image has zero annihilator is a maximal but not the largest radical class consisting of commutative idempotent rings.

On fuzzy subgroups

Abstract In this communication, we prove that (N(G), °) where N(G) is the set of all fuzzy normal subgroups of G, is a commutative idempotent semisimple semigroup. Also we give a fuzzy analog of the following well known result: If G is a group, H, L and K are subgroups of G such that H ⊆ L ⊆ G and K ⊆ G, then L ∩ (HK) = H(L ∩ K).

Remarks on duality and semigroups

This paper discusses the duality theory for compact commutative idempotent semigroups and compact commutative inverse semigroups. The major difference from earlier work is the use of semicharacters which are measurable with respect to some measure on the semigroup. Duality theorems are proved using measures which give different dual spaces.

Self-injective semigroup rings for finite inverse semigroups

1. An identity for R(S) and the main theorem. Let S be an inverse semigroup, i.e. a regular semigroup in which idempotents commute. Let E be the set of idempotents in S. Then E is a commutative idempotent subsemigroup of S and each principal left (right) ideal of S has a unique idempotent generator [1, Theorem 1.17, p. 28]. Then R(S) has an identity if R(E) has one. If Z denotes the ring of integers and Z(E) has an identity, then R(E) has an identity. Note that E has a zero if I El is finite.