Geodesic
The partial many-sorted algebras of terms and formulas with fixed variables count
Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 2, pp. 339-362

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Terms and formulas, which are formal expressions in first and second order languages obtained by alphabets, operation symbols, and relation symbols, are used to study algebras and algebraic systems. In this paper, we introduce the notion of terms with fixed variables count. The partial many-sorted superposition operations of such terms and their partial many-sorted algebra satisfying clone axioms as weak identities are presented. We also extend our structures from algebras to algebraic systems via the concept of formulas with fixed variables count. Conditions for the set of such formulas to be closed under taking of superposition of formulas are determined. We construct the partial many-sorted algebra of formulas with fixed variables count and investigate its satisfaction by clone axioms. Finally, we prove that such partial structure is isomorphic to some Menger systems of the same rank of partial multiplace functions.
Keywords: partial many-sorted algebra, term, formula, partial operation, representation 
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Kumduang, Thodsaporn; Leeratanavalee, Sorasak. The partial many-sorted algebras of terms and formulas with fixed variables count. Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 2, pp. 339-362. http://geodesic.mathdoc.fr/item/DMGAA_2023_43_2_a11/