Green’s Relations on Regular Elements of Semigroup of Relational Hypersubstitutions for Algebraic Systems of Type ((m), (n))

Any relational hypersubstitution for algebraic systems of type (τ,τ′) = ((mi)i∈I,(nj)j∈J) is a mapping which maps any mi-ary operation symbol to an mi-ary term and maps any nj - ary relational symbol to an nj-ary relational term preserving arities, where I,J are indexed sets. Some algebraic properties of the monoid of all relational hypersubstitutions for algebraic systems of a special type, especially the characterization of its order and the set of all regular elements, were first studied by Phusanga and Koppitz[13] in 2018. In this paper, we study the Green’srelationsontheregularpartofthismonoidofaparticulartype(τ,τ′) = ((m),(n)), where m, n ≥ 2.

Konsepsi siswa kelas tiga sekolah dasar tentang bilangan bulat

Artikel ini menggambarkan konsepsi salah seorang siswa kelas 3 Sekolah Dasar di Kota Bengkulu tentang bilangan bulat di luar pembelajaran formal. Siswa ini mengalami banyak intervensi tanpa skenario dalam pembelajarannya, di antaranya dari keluarga (kakak kelas 8 yang memiliki prestasi baik dalam matematika, kedua orang tua pengajar matematika) dan pelatihan sempoa. Konsepsi yang digali adalah arti bilangan negatif, bilangan bulat, serta operasi penjumlahan dan pengurangan pada bilangan bulat. Jenis penelitian ini adalah penelitian kualitatif dengan pendekatan studi kasus. Subjek diberikan beberapa pertanyaan seputar konsepsi bilangan bulat. Hasil penelitian menunjukkan bahwa subjek memiliki konsepsi sebagai berikut: 1) menggunakan istilah “kurang”, “utang” atau “posisi di bawah permukaan” untuk memaknai bilangan bulat ne­gatif, 2) bilangan bulat negatif diartikan sebagai invers penjumlahan bilangan asli, 3) ada perbe­daan antara simbol negatif dengan simbol operasi pengurangan, 4) bilangan bulat negatif bera­da di sebelah kiri bilangan 0 pada garis bilangan, 5) bilangan bulat negatif terkecil berada di sebe­lah kiri bilangan bulat negatif lainnya, seperti bilangan-bilangan pada penggaris, dan mampu menggunakan dinding sebagai pengganti garis bilangan, 6) menggunakan istilah “maju” atau “mundur” untuk mengoperasikan penjumlahan bilangan bulat, 7) menggunakan kata “jarak”, “lompatan di atas garis bilangan”, dan “lompatan di bawah garis bilangan” untuk mengoperasi­kan pengurangan bilangan bulat. Conceptions of third-grader elementary school about integersAbstractThis article described the conception of a third-grader elementary school in the City of Bengkulu about integers outside formal learning. This student experienced many interventions without scenarios in their learning, including their families (a brother in eighth-graders who had good mathematics achievements, parents were mathematics education lecturer) and an abacus trai­ning. The explored conceptions were the meaning of negative numbers, integers, and addition and subtraction operations on integers. This research was qualitative research with a case study approach. The subject was asked several questions regarding the conception of integers. The results of this research indicated that the subject had the following conception: 1) using the term “less”, “debt”, or “position under the surface” to interpret negative integers; 2) interpreting nega­tive integers as the inverse of the addition of natural numbers; 3) differentiating the negative symbol and the subtraction operation symbol, 4) locating negative integers to the left of “0” on a number line; 5) locating smaller negative integers to the left of other negative integers like num­bers on a ruler and having an ability to use a wall as a substitute of a number line; 6) using terms “forward” or “backward” to operate integer additions; and 7) using the term “distance”, “jumps over the number line”, and “jumps under the number line” to operate integer subtractions.

All intra-regular generalized hypersubstitutions of type (2)

A generalized hypersubstitution of type τ maps each operation symbol of the type to a term of the type, and can be extended to a mapping defined on the set of all terms of this type. The set of all such generalized hypersubstitutions forms a monoid. An element a of a semigroup S is intra-regular if there is b ∈ S such that a = baab. In this paper, we determine the set of all intra-regular elements of this monoid for type τ = (2).

Deterministic and non-deterministic hypersubstitutions for algebraic systems

Hypersubstitutions for algebraic systems are mappings which send operation symbols to terms and relational symbols to formulas preserving arities (see [D. Phusanga, Derived Algebraic Systems, Ph.D. thesis, Potsdam (2013)]). In the non-deterministic case, i.e. if one operation symbol is sent to several terms of the same arity and also one relational symbol is sent to several quantifier free formulas of the same arity, we can consider a mapping from the set of operation symbols into the power set of the set of all terms and from the set of relational symbols into the power set of the set of all quantifier free formulas of the considered type. These mappings are called non-deterministic hypersubstitutions for algebraic systems. We consider sets of algebraic systems which are invariant under non-deterministic hypersubstitutions and apply the result to [Formula: see text]-[Formula: see text]-solid classes of algebraic systems. In this paper, we consider an extension of non-deterministic hypersubstitutions which is based on deterministic ones.

Quadratic level quasigroup equations with four variables II: The Lattice of varieties

We consider a class of quasigroup identities (with one operation symbol) of the form x1x2?x3x4=x5x6?x7x8 and with xi?{x, y, u, v} (1? i ? 8) with each of x, y, u, v occurring exactly twice in the identity. There are 105 such identities. They generate 26 quasigroup varieties. The lattice of these varieties is given.

Green's Relations on HypG(2)

A generalized hypersubstitution of type τ = (2) is a mapping which maps the binary operation symbol f to a term σ(f) which does not necessarily preserve the arity. Any such σ can be inductively extended to a map σ̂ on the set of all terms of type τ = (2), and any two such extensions can be composed in a natural way. Thus, the set HypG(2) of all generalized hypersubstitutions of type τ = (2) forms a monoid. Green's relations on the monoid of all hypersubstitutions of type τ = (2) were studied by K. Denecke and Sh.L. Wismath. In this paper we describe the classes of generalized hypersubstitutions of type τ = (2) under Green's relations.

Nd-hypersubstitutions of many-sorted algebras

A non-deterministic hypersubstitution maps any operation symbol of type

Green's Relations on the Monoid of Regular Hypersubstitutions

The theory of hyperidentities and hypervarieties is based on the fact that the set Hyp (τ) of all hypersubstitutions of a fixed type τ forms a monoid, with a Galois connection between submonoids of this monoid and complete sublattices of the lattice of all varieties of type τ. For this reason, there is interest in studying the semigroup or monoid properties of Hyp (τ) and its submonoids. One approach is to study the five relations known as Green's relations definable on any semigroup. In this paper, we consider the type τ = (n) with one n-ary operation symbol for n≥ 1, and the submonoid Reg (n) of regular hypersubstitutions. We characterize Green's relations on every subsemigroup of Reg (n); then using this characterization we describe which subsemigroups of Reg (n) are 𝒢-subsemigroups of Reg (n) defined by Levi.

RELATING TREE SERIES TRANSDUCERS AND WEIGHTED TREE AUTOMATA

Bottom-up tree series transducers (tst) over the semiring [Formula: see text] are implemented with the help of bottom-up weighted tree automata (wta) over an extension of [Formula: see text]. Therefore bottom-up [Formula: see text]-weighted tree automata ([Formula: see text]-wta) with [Formula: see text] a distributive Ω-algebra are introduced. A [Formula: see text]-wta is essentially a wta but uses as transition weight an operation symbol of the Ω-algebra [Formula: see text] instead of a semiring element. The given tst is implemented with the help of a [Formula: see text]-wta, essentially showing that [Formula: see text]-wta are a joint generalization of tst (using IO-substitution) and wta. Then a semiring and a wta are constructed such that the wta computes a formal representation of the semantics of the [Formula: see text]-wta. The applicability of the obtained presentation result is demonstrated by deriving a pumping lemma for deterministic finite [Formula: see text]-wta from a known pumping lemma for deterministic finite wta. Finally, it is observed that the known decidability results for emptiness cannot be applied to obtain decidability of emptiness for finite [Formula: see text]-wta. Thus with help of a weaker version of the derived pumping lemma, decidability of the emptiness problem for finite [Formula: see text]-wta is shown under mild conditions on [Formula: see text].