Semigroup Forum最新文献

Monoids generated by elements of order two appear in numerous places in the literature. For example, Coxeter reflection groups in geometry, Kuratowski monoids in topology, various monoids generated by regular operations in language theory and so on. In order to initiate a classification of these monoids, we are interested in the subproblem of monoids, called strict Projection Involution Monoids (2-PIMs), generated by an involution and an idempotent. In this case we show, when the monoid is finite, that it is generated by a single equation (in addition to the two defining the involution and the idempotent). We then describe the exact possible forms of this equation and classify them. We recover Kuratowski’s theorem as a special case of our study.

Recently, Izhakian and Merlet gave a faithful representation (widetilde{rho }) of the Chinese monoid (Ch_{n}) of every finite rank n as a submonoid of the monoid (UT_{2cdot 3^{n-2}}(mathbb {T})) of upper triangular matrices over the tropical semiring (mathbb {T}). We exhibit another faithful representation (widetilde{phi }_n) of (Ch_{n}) as a submonoid of the monoid (UT_{n(n-1)}(mathbb {T})) of upper triangular matrices over (mathbb {T}). The dimension of (widetilde{phi }_n) is smaller than that of (widetilde{rho }) when (ngeqslant 4). Further, we give a faithful representation of the Chinese monoid ((Ch_n,~^sharp )) under Schützenberger’s involution (^sharp ).

In the theory of nonadditive integrals, an indispensable step is to define a pair of pseudo-addition and pseudo-multiplication that fulfill the conditional distributivity, leading to a structure of an ordered semiring in some sense. In this paper, we focus on conditionally distributive uninorms locally internal on the boundary, show that the second involved uninorm must be locally internal, and present a general framework of structures of such a pair of uninorms.

Some filter relative notions of size, (left( mathscr {F},mathscr {G}right) )-syndeticity and piecewise (mathscr {F} )-syndeticity, were defined and applied with clarity and focus by Shuungula, Zelenyuk and Zelenyuk (Semigroup Forum 79: 531–539, 2009). These notions are generalizations of the well studied notions of syndeticity and piecewise syndeticity. Since then, there has been an effort to develop the theory around the algebraic structure of the Stone–Čech compactification so that it encompasses these new generalizations. We prove one direction of a characterization of piecewise (mathscr {F})-syndetic sets. This completes the characterization, as the other direction was proved by Christopherson and Johnson (Semigroup Forum 104: 28–44, 2021).

We give a presentation for the monoid (mathscr{I}mathscr{O}_n) of all order-preserving transformations of an n-chain whose ranges are intervals. We also consider the submonoid (mathscr{I}mathscr{O}_n^-) of (mathscr{I}mathscr{O}_n) consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.

Let S be a discrete semigroup and let (^SS) denote the collection of all functions (f:Srightarrow S). If ((P,circ )) is a subsemigroup of (^SS) by composition operation, then P induces a natural topological dynamical system. In fact, ((beta S,{T_f}_{fin P})) is a topological dynamical system, where (beta S) is the Stone–Čech compactification of S, (xmapsto T_f(x)=f^beta (x):beta Srightarrow beta S) and (f^beta ) is a unique continuous22 extension of f. In this paper, we concentrate on the dynamical system ((beta S,{T_f}_{fin P})), when S is an arbitrary discrete semigroup and P is a subsemigroup of (^SS) and obtain some relations between subsets of S and subsystems of (beta S) with respect to P. As a consequence, we prove that if ((S,+)) is an infinite commutative discrete semigroup and (mathcal {C}) is a finite partition of S, then for every finite number of arbitrary homomorphisms (g_1,dots ,g_l:mathbb {N}rightarrow S), there exist an infinite subset B of the natural numbers and (Cin mathcal {C}) such that for every finite summations (n_1,dots , n_k) of B there exists (sin S) such that

We utilise directed graphs called isoterm graphs to show that the variety generated by M(abba) has continuum many subvarieties.

In our earlier article (Can and Sakran in Port Math 81(1–2): 21–55, 2024) we initiated a study of the complement-finite submonoids of the group of integer points of a unipotent linear algebraic group. In the present article, we continue to develop tools and techniques for analyzing our monoids. In particular, we initiate a theory of ideals for unipotent numerical monoids.

Let S be a semigroup, Z(S) the center of S. In this paper, we determine the complex-valued solutions of Kannappan–d’Alembert’s functional equation

and Kannappan–Wilson’s functional equation

where (mu ) is a measure that is a linear combination of Dirac measures ((delta _{z_i})_{iin I}), such that (z_iin Z(S)) for all (iin I), and (sigma :Srightarrow S) is an involutive automorphism or an involutive anti-automorphism for the first equation and an involutive automorphism for the second one. We also give some interesting applications.

This paper focuses on the study of H-closedness and absolute H-closedness of the posets. First, we propose a counterexample to indicate that an absolutely H-closed topological semilattice may not be c-complete, which gives a negative answer to an open question proposed by Banakh and Bardyla. However, in the case of continuous semilattice with the Lawson topology, we prove that the absolutely H-closed topological semilattice implies c-completeness. Second, we obtain a characterization for quasicontinuous lattices utilizing the topological embedding mapping. Finally, enlightened by the definitions of H-closedness for Hausdorff spaces and absolute H-closedness for Hausdorff topological semilattices, we introduce the concepts of H-closedness and absolute H-closedness for posets with the Lawson topology.